The post All The Logarithm Rules You Know and Don’t Know About appeared first on Math Academy.

]]>**Table of content:**

- Common logarithm
- Natural logarithm
- Logarithm Rules, identities, and formulas
- Some solved examples
- Characteristic & Mantissa
- Number of digits
- Things to remember
- Antilogarithm
- Cologarithm
- Harmonic logarithm
- Logarithmic Series
- False Logarithmic Series
- Logarithm quotes

All you need to know is that logarithms are essentially a different format for writing exponents. If you know how to use exponents and you know the format for logarithms, then you’ll do fine in this section.

Let’s take an example:

If log_{4}64 = x, then what is the value of x?

Virtually every major test contains some variation on this problem, so plan ahead to make sure you know how to answer this problem type.

You’ll be given a logarithm problem with one value (x, y, or z) missing. All you need to do is rewrite that sucker as an exponential statement. Take a look!

We’re given: log_{4}64 = x

Rewritten: 4^{x} = 64

That’s a lot easier to work with, isn’t it? Find matching bases and you’re practically done!

4^{x} = 4^{3}

x = 3

We call “Common logarithm” the logarithm function with the base 10. Generally noted by “” or “”. It is also known as the “decadic logarithm” and as the “Decimal logarithm”, and also known as the “Briggsian logarithm” in honor of the British mathematician Henry Briggs how was notable for changing the original logarithms invented by John Napier into the logarithm with base 10.

The common logarithm of is the power to which would have to be raised to equal , meaning that common logarithm is the inverse of the exponential function .

Graph of the common logarithm

While the common logarithm has a base 10, the Natural logarithm has a base , where is the mathematical constant known as “Euler’s number” or “Napier constant”; it is an irrational and transcendental number, its value is approximately , this constant is also widely used as a base for exponential functions. Natural logarithm is noted or . The natural logarithm of is the power to which the constant would have to be raised to equal , meaning that the natural logarithm is the inverse of the exponential function .

Here is Graph of the natural logarithm and the exponential function . Note the natural logarithm and the exponential function base are symmetrical with respect to the line .

Common and natural logarithms can be expressed in terms of each other as follow:

1- Logarithm of the base: because

2- Logarithm of 1: because

**Logarithm identities for canceling exponentials:**

3- because

4- because

**Logarithm identities for the different operations:**

5- Logarithm of product: because

6- Logarithm of the reciprocal:

7- Logarithm of a quotient: because

8- Logarithm of the reciprocal of a power of base:

9- Logarithm of a power:

10- Logarithm of a square root:

11- Logarithm of the n root:

12- Logarithm to a quotient base:

13- Logarithm to a power base:

14- Logarithm of a power to a power base:

15- Power with an logarithmic exponent to the same base as the power:

16- Power with a logarithmic exponent: because

17- Since the logarithm is a monotonic function, more precisely monotone increasing function if the base and monotone decreasing function if the base , therefore we have:

18- Logarithm with the base is the Natural Logarithm

19- Logarithm transitive property:

20-

21-

22-

23- For are real positive numbers and and and are real numbers; because

24-

25-

**Changing the logarithm base:**

26- Changing the logarithm base:

This identity of changing the base is very useful to calculate the wanted logarithm using the calculators, espacealy useful since nearly all calculators have a “” and an “” buttons, but not all of the calculators have a button for a logarithm with a random base.

27- The formula of base changing has many consequences, from which we cite:

Where is any permutation of the subscript . For example:

28- Changing the natural logarithm (base ) to the common logarithm (base 10):

29- Switching the logarithm base with the logarithm argument:

30- From the previous one we can write also: (Canceling logarithm by multiplying by the logarithm with the base and the argument switched)

**Summation and subtraction identities:** This two identities are especially useful in probability theory, when dealing with logarithmic probabilities:

31- because:

32- because: with (or else the subtraction identity won’t be defined since the logarithm of 0 doesn’t exist).

33- More generally:

**Some useful exponent identities:**

34-

35-

**Some other identities:**

36-

37-

**Common mistakes that you should avoid and be careful from:**

- The log of a sum isn’t the sum of the logs, because the sum of the logs is the log of the product, and the log of a sum can’t be simplified.

- Also, the log of a sum isn’t the product of the logs, it is tricky if you are not careful since it is close to the logarithm product rule.

- The product of the logs isn’t the log of the product.

- The log of a difference isn’t the difference of the logs, the difference of the logs is the log of the quotient, and the log of the difference same as the log of the sum can’t be expressed in a simpler way.

- Also, the log of the difference isn’t the quotient of the logs, its the opposite that is true, the log of a quotient is the difference of the logs.

- The exponent to which the log is raised to a power of, isn’t the coefficient of the log; the logarithm raised to a power can’t be simplified.

- The log of a quotient isn’t the quotient of the logs.

**Limits of the logarithm function:**

38- Logarithm of 0: is undefined;

We have:

39- when tend to infinity:

40-

41- ;

this limit is usually summarized and made easy to memorize by the expression “Logarithms grow more slowly than any power or root of “.

**Derivatives of logarithm functions:**

With , and , We have:

42-

43-

**Integrals of logarithmic function:**

44- Integral definition:

45-

46-

47- For higher-order integrals, it is convenient to define: , where is the n harmonic number.

and so we get:

Therefore:

**َAbsolute value:**

When dealing the logarithm, we need to keep an eye on the signs and the values of the base and the argument of the logarithm function, we can summarize the conditions we have to verify as follow:

We have the definition of logarithm:

- The base can’t equal 1,
- The base can’t be negative,
- The argument of the logarithm function can’t be less or equal to 0.

These conditions must be taken into consideration when with real numbers, and to do so we use the absolute value when necessary to ensure that the conditions are always met.

In the case of real numbers, we introduce the absolute value to some of the logarithm rules, as follow:

48-

49-

50-

**Complex logarithm:**

51- The logarithm to the real base of a negative number is undefined.

For complex number in the polar form:

Then the complex logarithm for negative is given by:

Taking in account periodicity we can write:

we find:

52-

53-

54-

55- for

56- for

57- for

58- for

59- for and

60-

61- (mod )

62- with

63- with

64- with

65- with

Why not take a look at the 3D plot and the contour plot of the logarithm and the exponential functions!!!

**Logarithmic inequalities:**

66- If

67- If

68- If

69- If

70- If

71- If

72- If

73-

74-

75-

Here is a summary of the main rules of the logarithm

- If 64 = x
^{2}then log_{x}64 = ?

Here, x = 8

So, log_{8}64 = log_{8}8^{2} = 2

- 2 log
_{2}4 + log_{2}x =-1, then x = ?

Here, log_{2}4^{2} + log_{2}x = -1

log2 ( 16x) = -1

16 x = 2^{-1}

16x = ½

x = 1/32

- log
_{216}y= 1/3, what is the value of y ?

y = 216^{1/3}

^{ }y=6

- If log
_{x}9 – log_{x}4 = 1/2, then x = ?

log_{x} (9/4) = ½

9/4 = x1^{/2}

Squaring both sides,

x = 81/16

- 8
^{3}= 512 - 7
^{2}= 49 - 6
^{x}= 216

(*Answers at the bottom of the article)*

- log
_{3}81 = x, then x = ? - If log
_{x}196 = 2, then x = ? - If log
_{2}(x) = 5, then x = ?

(*Answers at the bottom of the article)*

The logarithm of a number has an integral part and a decimal part. The integral part is called the characteristic and the decimal part is called the mantissa.

For example, log_{10} 91000

= log_{10} (10^{4} x 9.1)

log_{10} 10^{4} + log_{10} 9.1

= 4 + 0.959

Here, 4 is the characteristic and 0.959 is the mantissa.

A significant property of the characteristic (for common logarithms) is that (characteristic + 1) will give the number of digits of the integral part of the number whose logarithm you determined. In the above example, 91000 contains (4 + 1) = 5 digits.

This is explained as follows:

log_{10} 91000 = 4.959,

91000 = 10^{4.959} = 10^{4+ 0.959} = 10^{4} 10^{0.959}

= 10* ^{x}* 10

Since *y* will always be < 1, 10* ^{y}* will always be < 10. Hence, the number of digits in the number will be equal to the number of digits in 10

Also, the characteristic of a logarithm could be either positive or negative; however, the mantissa should always be positive.

For example, log_{10} 0.091 = log_{10} (10^{–2} 9.1)

–2 + 0.959 =

If the calculated value of the logarithm of some number is negative (usually, calculators generate values containing a negative mantissa), then we make the mantissa positive as follows:

–1.041 = –(1 + 0.041) = –1 – 0.041

= (–1 – 1) + (–0.041 + 1)

= –2 + 0.959

We call “Antilogarithm” the inverse of the logarithm function, and we write as follow:

, in other words and to put it simply the antilogarithm is the exponentiation! For example the antilogarithm of is , and the antilogarithm of is .

Note that since each function is the inverse of the other then we have the .

For example, the antilogarithm of in base is

We call “Cologarithm” the logarithm of the reciprocal of a number , which is, using the logarithm properties, equal to the negative of the logarithm of this number, we can write as follow:

We call “Harmonic logarithms” of order and degree , the unique functions that, for all integer () and for all non-negative integer (), verify the following:

- ,
- has no constant term except ,
- ,

Where the “Roman symbol” is defined by

This gives the special cases:

Where is a Harmonic Number:

The harmonic logarithm has the Integral

The harmonic logarithm can be written

Where is the Differential Operator, (so is the integral). Rearranging gives

This formulation gives an analog of the Binomial Theorem called the Logarithmic Binomial Formula. Another expression for the harmonic logarithm is

Where is a Pochhammer Symbol and is a two-index Harmonic Number.

**References**

- Loeb, D. and Rota, G.-C. “Formal Power Series of Logarithmic Type.” Advances Math. 75, 1-118, 1989.
- Roman, S. “The Logarithmic Binomial Formula.” Amer. Math. Monthly 99, 641-648, 1992.

We can express and expand some logarithm expressions as a series of powers of , if we have

Let’s take a look at some logarithm series

- We can write an expansion of if as follow:

Then we can deduce the expansion of by replacing with and we will get:

Or in other way:

Now let’s take a look at some results derived from the previous series using the properties of the logarithmic function

- The previous algorithmic series isn’t valid if but what if ? Well, it is proved that logarithmic series is valid also for , and we get the following:

Note that the logarithm series isn’t valid when and it doesn’t have a sum, and this makes sense since isn’t a finite quantity.

- If we look at the differences between the exponential series and in comparison with the logarithmic series we find that the exponential series is expressed by the sum of positive terms (positive signs) but the terms of the logarithmic series are alternatively positive and negative.

Also the logarithmic series is valid only when whereas the exponential series is valid for every value of .

- Some examples of Infinite series of various simple functions of the logarithm:

Where represent the Euler-Mascheroni constant and is the Riemann zeta function.

We call False logarithmic series the function expressed as follow:

This function is interesting since it gives the same image of the logarithmic function for every natural number, i.e. , or in other terms the graph of the false logarithm function intersect with the graph of the logarithm function for every such that $x\in\mathbb{N^{*}}.

Here is a graphical representation of the false logarithm function and the logarithm function, notice the intersection points of the two graphs.

- Logarithms are generally expressed to the base 10. These are called common logarithms. If no base is mentioned, it is assumed that the base is 10.
- Logarithms are not defined for zero or negative numbers. They are defined only for positive numbers.
- Logarithms can be expressed in any base.
- Logarithms expressed in one base can be converted to logarithms expressed in any other base.

The spectacular thing about Johnny [von Neumann] was not his power as a mathematician, which was great, or his insight and his clarity, but his rapidity; he was very, very fast. And like the modern computer, which no longer bothers to retrieve the logarithm of 11 from its memory (but, instead, computes the logarithm of 11 each time it is needed), Johnny didn’t bother to remember things. He computed them. You asked him a question, and if he didn’t know the answer, he thought for three seconds and would produce an answer. — Paul R. Halmos

Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter. — Augustus de Morgan

You have no idea, how much poetry there is in the calculation of a table of logarithms! — Carl Friedrich Gauss

*Answers:*

*8*^{3}= 512 ; log_{8}512 = 3*7*^{2}= 49 ; log_{7}49 = 2*6*^{x}= 216 ; log_{6}216 = x*log*_{3}81 = x, then x = ? 4*If log*_{x}196 = 2, then x = ? 14*If log*_{2}(x) = 5, then x = ? 32

Well, here we arrive at the end of this journey in this logarithmic paradise if you have anything you want to be added to this article just let us know, and we will make sure to add it to make this article the ultimate logarithmic article. Also, make sure to check the blog post about Absolute Value and Logarithms.

And don’t forget to join us on our Facebook page stay updated on any new articles and a lot more!!!!!

The post All The Logarithm Rules You Know and Don’t Know About appeared first on Math Academy.

]]>The story of integration started with I. Newton, when in the late 1660 he invented the method of inverse tangents to find areas under curves.

In 1680, G. Leibnitz discovered the process of finding tangent line to find area. Thus, they had discovered the integration, being a process of summation, was inverse to the operation of differentiation since finding tangent lines involved differences and finding areas involved summations.

In 1854, G.F Riemann formulated a new and different approach to define integral on the real line. He separated the concept from its differentiation. His approach was to examine the motivating summation and limit process of finding areas by itself.

In 1875, J.G Darboux viewed Riemann Integration in a different way. The approaches of both Riemann and Darboux demanded the integration to be bounded in its domain. It has been established that the two definitions of definite integral given by Riemann and Darboux are equivalent that is why Riemann integral are often called Darboux-Riemann Integrals.

In this article we will discuss about the integral based on Riemann and Darboux-Riemann approach will be discussed in mainly on real line ().

__Definition__:

__Partition__: Let be a given closed and bounded on real line. Let be finite number of points in such that , then the finite order set

is defined to be a partition of which divides into “” closed sub-interval , for all .

__Fact__: Every partition “” of must include at least two points “” and “” of . In fact, is the trivial partition of .

__Definition:__

__Family of partition__: Every bounded and closed interval has infinite numbers of partitions. The set or the collection of all partitions of , called family of partition of , is denoted by .Thus, if implies “” is a partition of .

__Example:__

For positive integer “”,

is a partition of

is a partition of

is not a partition of as .

__Definition__:

__Norm of partition__:

If be a partition of and denotes the sub-intervals of (for ) having the length , for .

Then the greatest length of the sub-intervals i,e., is defined to be a norm of the partition “” and denoted by or .

i,e., .

__Definition__:

__Upper Sum__: Let, be a bounded function on and be a partition over .

Then “” is bounded on every .

Let,

Then,

defined as upper sum or Darboux upper sum._{ r}

__Lower sum:__ Let, be a bounded function on and be a partition over .

Let,

Then,

defined as lower sum or Darboux lower sum.

_{r}

__Notation:__

__Upper Sum__:

For the function “” over the partition”” on ,

Then,

__Lower Sum__: For the function “” over the partition “” on ,

Then,

Facts:

- If and are equal when is a constant function.

2.

is the ‘oscillatory sum’ of “” for the partition “” on , and we write that as

- For any bounded function “” on and ,

Where, , are infimum and supremum of “” on .

__Definition__

__Refinement of a partition__:

Let, and be two partitions of such that if then is called refinement of the partition “”.

__Definition__:

__Common Refinement:__ If and are two partitions of then and .

So is the refinement of both and and is called common refinement of and .

__Properties of refinement of a partition:__

If “” is bounded on and where “” is the refinement of “”

Then

Let “” be a real valued function over , then is defined as lower (Darboux) integral of “” on and denoted by

__Darboux Upper Integral__:

For real valued function “” on , then is defined as upper (Darboux) integral of “” on and denoted by

__Particular fact__:

For any real valued bounded function on and any two partitions , of

__Definition__:

A real valued function “” on is said to be Riemann integrable on if

Then “” is Riemann integrable and are denoted that integral by

and when “” is Riemann integrable we denote that as

__Example__:

Let

Check

__Solution__: Let

be a partition of where “” is a positive integer.

Then “” divides into “” sub-intervals

Here is monotonically increasing and continuous function on

Also

So,

And

Here,

Hence,

And

So, we understand by the example how we can use the definition of Riemann Integration.

But for more complicated functions there will be difficulties to find maximum and minimum in every subinterval so, now we are going to modify the theory of “when a function is said to be a Riemann integral.”

__Theorem :__ let be a bounded function on . A necessary and sufficient condition for inerrability of an is that for every position such that

__Proof : __ (Necessary Part:)

Let be integrable on .

Then

Let be any positive number. From the definition of lower and upper integral of on , we have for the given positive , there exist a partition on such that

Also

Therefore

Hence

Conversely (Sufficient condition)

Suppose for any positive number there exists a partition on such that

Since,

and

__A__lso

Therefore

Hence

Therefore

Hence is Riemann Integrable on

Hence proved.

__Remark:-__

let be a bounded function on . If be a sequence if partitions of such that the sequence converges to zero and

Then is Riemann integrable on .

Now we are going to learn how to use the Remark

Example : let be a function by

Here we are going to check the integrability of

__Lets try:__ Here is bounded on Let for we choose a partition

s

Here contains rational as well as irrational points for each

So,

Then,

Also

Then

Hence by the Remark we can say that is not Riemann integral on .

- If is monotonic then . i.e is Riemann integrable.

- If is continuous then is Riemann integrable.

- If be bounded but has finite number of discontinuous points on then

- If bounded but has infinite point on discontinuities in such that the number of limits Of these infinite discontinuous points is finite in then .

- If is integrable on , then is integrable on every closed sub interval of .

- If , and is integrable on and on then

- If is integrable on then for every then is also integrable on and further more,

- If two integrable on and

- If and then for any integrable on furthermore,

(x) If is integrable on , then on integrable. But the converse is not true.

(Why?)

Here, is not Riemann integrable but

is constant then it must be Riemann integrable on .

- If and are both integrable on , then also integrable on .

- If is integrable on , these is integrable on but the converse is not true (try with the same function

- If and integrable and where , then also integrable as .

- If is Riemann integrable on and where then also integrable on .

- If and bounded and closed interval. Let and integrable and continuous function such that then is integrable on .

Example 1 : – let be a bounded function on defined by

Now we are going to check is Riemann Integrable or not.

Let’s start:- Here is bounded function also is continuous on except , so is Riemann Integrable on .

But

As , , so is not bounded on and hence is not Riemann Integrable on and we are done.

Example 2 Consider the function such that

Here also we are going to check integrable or not in and if integrable then we will find the value of the integration.

Let’s start

Here the expression of is

Since for all in , is bounded on . is continuous on except the points and . So has finite number of discontinuity on then we can say that is integrable on

So hence is Riemann Integrable on

Now we are going to find the value of the integration of the function on .

Let we define

And then becomes

Hence

Therefore we can do this now

Therefore,

And we are done with this problem

Now try this problem with same manner such that

Try to prove is Riemann Integrable and find (Answer )

- If is integrable on and for all values of in .

Then

- If and is integrable on and for all values of in .

Then

- If is integrable on

Then

__Problems related with these theories:-__

__ ____Example.__

Show that

To prove this we are going to use all of the theories which we learnt till now,

Let

Here for all

For all

For all

Therefore let for all

and

Here

for all

So therefore

Now

`

And

Therefore

Hence proved.

Now some work sheet problems:-

- Let

Check is Riemann Integrable or not.

- Try to prove these problems

i)

- ii)

iii)

- Try to show that

__ __

- Elementary Analysis: The theory of Calculus; Kenneth Ross.
- Improper Riemann Integration; Markos Roussos
- Modern Theories of Integration; H. Kestelman
- The Riemann Approach to Integration; Washek Pfeffer.

__ __

Thank you………

The post Riemann Integral appeared first on Math Academy.

]]>The Riemann integral of a bounded function over a closed, bounded interval is defined using approximations of the function that are associated with partitions of its domain into finite collections of subintervals. The generalization of the Riemann integral to the Lebesgue integral will be achieved by using approximations of the function that are associated with decompositions of its domain into finite collections of sets which we call Lebesgue measurable. Each interval is Lebesgue measurable. The richness of the collection of Lebesgue measurable sets provides better upper and lower approximations of a function, and therefore of its integral, than are possible by just employing intervals. This leads to a larger class of functions that are Lebesgue integrable over very general domains and an integral that has better properties. For instance, under quite general circumstances we will prove that if a sequence of functions converges point-wise to a limiting function, then the integral of the limit function is the limit of the integrals of the approximating functions.

One of the primary motivations for developing a theory has to do with the failure of the theory of Riemann integration to behave nicely undertaking point-wise limits of functions. In particular, it is well-known that point-wise convergence of a sequence of functions does not translate to the interchange of limits and integrals; for example, if

then as , converges point wise to the zero function. If we take its integral,

Which leads to

But

This inability to interchange a limit and an integral can be extremely inconvenient. One can interchange a limit and Riemann integration under certain circumstances like uniform convergence, but this is usually far too restrictive of a condition. What measure-theoretic integration gives you is a way to do this interchange provided us, have point-wise convergence and much weaker conditions than uniform convergence; in exchange, it forces us to throw out sets of measure zero. In addition, once the theory is developed we suddenly have extremely powerful tools to take limits of integrals you did not have before (Lebesgue dominated convergence theorem).

So we have two main motivations to make a rich theory

- To integrate more functions (Like Dirichlet Function )
- To make a space this is completely integrable.

There are more motivations like

- Interchanging limits and integrals (e.g. the limit of a sequence of continuous functions may not be Riemann integrable)
- Length of Curves (finding the length of curves that are only rectifiable, not continuously differentiable), etc.

To define Lebesgue Measure on a set we need some precious definitions and some important functions and classes or collection of subsets of .

Let denote the collection of all intervals of . If an interval has

end points and we write it as . By convention, the open interval

. Let .

Define the function by

This is called length function defined on .

- is monotonic i.e.
- is finitely additive.

Let be such that where for Then

- is finitely sub-additive.

Let be such that Then

- is countable additive .

Let be such that where for Then

- is countable sub-additive .

Let be such that where

Then

- Translation Invariance

For every and .

Let be any non-empty set and let a collection of subsets of . Then the collection is called an algebra of subsets of if satisfies following properties:

- whenever
- whenever

Then is called Algebra on .

Let be any non-empty set and let a collection of subsets of . Then the collection is called an Algebra of subsets of if satisfies following properties:-

- whenever
- for

Then is called Algebra on .

Def:- For every subsets of , the Lebesgue Outer Measure of , denoted by is defined by

Where varies over all possible sequence of open intervals of whose union contains .

The Lebesgue Outer Measure is generated by length function which is defined on earlier so it’s preserves some of their properties.

- is monotonic i.e.
- is finitely sub-additive .

Let be such that Then

- is countable sub-additive .

Let be such that where

Then

- Translation Invariance

For every and .

Now we already noticed that hasn’t had the properties of finitely additive and not countable additive (why?)

A Vitali* Set in , has a positive measure

Specifically, let denote translation. (\textit{That is, for and let }.) Note that outer measure is invariant under translation, so

Now let be a Vitali set, and let be an enumeration of the rationals in . By construction of , the sets are pair wise disjoints and their union is . By countable sub-additively we have

In particular we must have

So we can find an integer sufficiently large that

,

Let .

Then the sets are pair wise disjoint, and since

We have

.

Hence by monotonicity, .

On the other hand,

So we have

So we saw that is not finitely additive and it is also not countable sub-additive.

__Vitali:__ An elementary example of a set of real numbers which is not Lebesgue measurable.

__Definition:__

Measurable Set(Carathéodory Condition): A set is said to be Lebesgue-Measurable if

This Condition is known as Carathéodory Condition.

- If is Lebesgue Measurable then also Lebesgue Measurable set in
- and are Lebesgue Measurable
- Every Interval in is Lebesgue Measurable.
- Every countable set is Lebesgue Measurable with measure zero i.e.

If is a countable set in then

- Every uncountable set has a non-zero measure (except Cantor Set)
- Every Borel* set is Lebesgue Measurable.

(Borel Set: Collection of all and sets)

Definition:

__Class :-__ Collection of all Lebesgue Measurable sets in .

This Collection is a Algebra and it is the largest measurable Algebra over and () is called Measure Space which we want to achive.

Now we got a new developed theory on subsets of and we can now use the theory to integrate those kinds of functions which we can’t integrate by using Riemann Integration. Also, we can use the theory to find the length of those kinds of curves which is not continuously differentiable on and we can use to find the integration over rationalize functions too. This rich theory gave us more freedom to apply on the space or subsets in .

- Yeh ; Real Analysis
- Robert G. Bartle; The Elements of Integration and Lebesgue Measure.
- Sheldon Axler; Measure, Integration, and Real Analysis.

Thank You.

Don’t forget to join us on our Facebook page for any new articles and a lot more!!!!!

The post Lebesgue Outer Measure appeared first on Math Academy.

]]>The post Permutations & Combinations made easy! appeared first on Math Academy.

]]>This lesson covers the topics of combinatorics (combinations, permutations with and without repetitions, circular and constrained permutations). It will equip you with all the necessary theoretical knowledge on the topic, and show how it is applied in standardized test problems.

The factorial of an integer n is defined as the successive product of all the positive integers from n down to 1:

n! = n × (n − 1) × (n − 2) × (n − 3) × . . . 1

By convention, 0! = 1.

0! and 1! are the only two factorials that are odd; the rest have a factor of 2 in them.

- 1! = 1
- 2! = 2 × 1 = 2
- 3! = 3 × 2 × 1 = 6
- 4! = 4 × 3 × 2 × 1 = 24
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
- 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

On most of the tests, you won’t need to calculate large factorials. Most of the time you will be able to reduce them or simplify the expression.

For example: Find the value of 23!/21!

- 21
- 23
- 462
- 506
- 10626

If you write down all the factors, you will see the trend:

(23 × 22 ×21 × 20 × 19 × 18…..1) / (22 × 21 × 20 × 19 × 18…1).

You can cancel all the numbers in the numerator against the same numbers in the denominator except the 23×22 in the numerator. So,

23!/21! = 23 × 22 = 506.

Combinations mean groupings of elements in which the order of the elements does not matter.

The main formula for combinations is

*where nC _{k} is the number of possible combinations of k elements, taken from the broader set of n elements.*

Example: A box contains seven cubes, each one of a different color. If you take two cubes out of the box, how many different color combinations are possible?

- 12
- 17
- 21
- 38
- 42

Here,

Hint: you can easily see that if you need to pick one element out of the set of n elements, there are n ways to do it. (E.g. pick one day out of 365 – there are 365 ways to pick it). If you need to pick two elements, the formula

This is just the same formula as above, only simplified for the case of two elements. You can remember it, or you can use the general formula, whichever is more convenient for you.

Permutations are ordered arrangements of elements. The number of possible ordered permutations is almost always larger than the number of unordered combinations of the same set of elements.

The formula for permutations is

*where nP _{k} is the number of possible ordered arrangements of k elements from a larger set of n elements.*

Example: There are seven cubes of different colors in a box. If you take two cubes out of the box, one with your right hand and another one with your left hand, and the place in one or the other hand is important, how many different combinations can be made?

- 12
- 17
- 21
- 38
- 42

Here,

These permutations are also called permutations without repetition.

Example: How many different ways can four doctors be assigned to four patients if each doctor gets one patient? The first doctor has four choices, the second doctor then has three choices, the third doctor has two choices, and the fourth has just one choice. Then, the total number of arrangements is

4×3×2×1 = 4! = 24.

If there are still 4 patients but more doctors, say 6, then you have permutations of a subset. Let’s find the number of ways 6 doctors can be assigned to 4 patients. The first patient can get any of the 6 doctors, so there are 6 options. The second patient can be treated by any of the 5 remaining doctors (since the same 6th doctor cannot have 2 patients), so there are 5 options. Then, there are 4 options to assign a doctor to the third patient and 3 options for the 4thpatient. The total number of options is the product of the options for each patient

= 6 × 5 × 4 × 3 = 360 ways.

Another example: What is the number of the possible three-letter arrangements of letters K, L, M, N.

There are 4 options for the first letter, 3 for the second, and 2 for the third,

So, there are 4 × 3 × 2 = 24 options in total.

If you need to rearrange a set that has repeating elements, you need to adjust the formula for these repetitions.

Example: In how many ways can the letters of the word SOLO be rearranged?

The formula to use is

*where n is the number of elements in the original set, in which there are j different repeating elements, and r is the number of times each element repeats.*

Returning to the SOLO example, there are 4 letters in total, so

n = 4,

and there is one repeating letter, so

j = 1.

Then, the number of repetitions of the letter O is 2, so

r1 = 2.

Plug this into the formula:

To take a broader view of permutations and combinations, you need to understand how they relate to each other. We have said that permutations are basically the same as combinations, but with the order of elements taken into account.

This is reflected by the extra element in the denominator in the combinations’ formula:

Here’s an example to clarify this.

You need to select four bands to perform at a concert from a total of 8 available bands, and you need to decide on the order in which they perform.

The usual way to do this is the same as in the above example with doctors and patients: there are 8 options for the first slot in the concert, 7 options for the second slot, 6 for the third and 5 for the last. This makes a total of 8 × 7 × 6 × 5 = 1680 possible options.

**Another way to do this is to apply the permutations formula:**

** **

Here, n = 8 since there are 8 bands in total, k = 4 since you need to pick 4. Then

**One more way of doing this is by breaking the problem down into two steps:**

First, you need to identify the four bands, and then you need to arrange them in a certain order. This is a less convenient way of solving this particular problem, but you need to be familiar with the logic, as you might need it for more complex problems.

The first step is to select 4 bands out of 8, not taking order into account. This is done with the combinations formula

*Where, again, **n=8 and k = 4.*

There are 4 × 3 × 2 × 1 = 24 ways to order this set. Now, you have 70 ways to select bands and 24 ways to order them. This makes 70 × 24 = 1680 ways to pick an ordered combination of bands. You have reached the same result. This should give you an idea of how combinations and permutations are related so that you could apply this logic to problems of any difficulty level.

In this problem type, objects are arranged around a circle. The difference as compared to linear permutations is that a circle does not have a clear beginning and end, like a line. So, although combinations ABCD and BCDA are two different options in a linear permutation, they are the same combination around a circle:

So, in a circle, only the distinct order of elements with respect to each other matters; it does not matter from which point on the circle we start counting them. We can start with A, resulting in ABCD, or we can start with C, resulting in CDAB – they are the same combination around a circle.

If you can arrange four elements in a line in 4! ways (4 options for the first place, 3 for the second, 2 for third, 1 for fourth

= 4 × 3 × 2 × 1 = 4! = 24,

there are fewer options for arranging the same four elements around a circle. We must exclude the identical combinations that were identified above. There are as many repeating combinations as there are slots around the circle. If ABCD is the same combination as BCDA, CDAB, and DABC, this makes four different linear options for everyone’s circular combination. (There are four different options to place the first element, A, without changing the order of the elements with respect to each other). So, in order to find the number of circular permutations, we should divide the number of linear permutations by the number of slots around the circle (or the number of possible ways to place the first element). So, the formula for circular permutations is

where *m *is the number of slots around the circle. In the case of four elements, there will be

Some permutations problems may have certain constraints, for example, a certain two elements cannot be placed next to each other. Let’s take an example: Given 5 elements to order, A, B, C, D, and E, with the constraint that elements A and E cannot be placed next to each other, how many ways are there to order the elements?

There are two ways to approach this. First is the classic way of multiplying the options: there are

5 options for placing the first element, A. (Let’s start with the one that is involved in the constraint.) This leaves either 3 or 2 options to place E (because if A is placed in the first or last place, there is only one place next to it that is not available for E; but if A is placed in one of the middle places, there are two places beside A that are closed for E).

There are two “edge” options for A, and three “middle” options for A. This means, that in 2/5 cases E will have 3 options, and in 3/5 cases E will have 2 options. After we’ve placed E, there are 3 options left for B, 2 for C, and 1 for D. Multiply the options:

5 × (2/5 × 3 + 3/5 × 2) × 3 × 2 × 1 = 72.

Now, there is another way to do this. Let’s count the total number of possible arrangements for 5 elements:

5! = 5 × 4 × 3 × 2 × 1 = 120.

Now, let’s exclude the options where A and E are placed together. There are 4 pairs of places that can be occupied by A and E, and 2 options per each place (AE or EA). This makes 8 options for A to be next to E.

If the two places for A and E are defined, there are 3! options to place the remaining elements B, C, and D. So the total number of combinations of 5 elements in which A and E are placed together, is

8 × 3! = 8 × 3 × 2 × 1 = 48.

Therefore, the total number of options to place 5 elements so that A and E are not placed together is

120 – 48 = 72.

You might encounter a problem where it is convenient to use combinatorics to calculate probability.

Consider this example: There are 6 different cubes in a box: yellow, red, green, blue, orange, and purple. What is the probability that if you pick 4 cubes out of the box, a yellow and a blue one will be among them? Here, it is convenient to use the probability formula:

The total number of 4-cube combinations is

The number of 4-cube combinations that contain a yellow cube and a blue cube is

We are looking for 4-cube combinations in which 2 cubes are already chosen: a yellow one and a blue one. We need to select the other 2 cubes from the remaining 4 cubes in the box, and there are 4C_{2} ways to do this.

Now calculate the probability:

P = 6/15 = 2/5.

- Use the formula for combinations when the order does not matter, and the formula for permutations when the order matters.
- Sometimes it is more convenient to calculate the probability of the event not occurring to answer the question.

The post Permutations & Combinations made easy! appeared first on Math Academy.

]]>The post Continuity – Calculus appeared first on Math Academy.

]]>**Table of content:**

- Introduction
- Continuity of a function
- Properties of continuity
- Discontinuity of a function
- Types of discontinuity
- The theorem of indeterminate value
- Conclusion

In the previous article, we learned about how to study a function, and their domains of definition and codomains, a since we know that the domain of definition of a function is the set of possible values for which the function has an image or in other terms, but this may push ask some questions: does the graph of every function is formed of a single part or a single curved line?! Also, what about composite functions?!! How would their graphs look like, and does it always come in a signal part?!! Well, these questions and more will be answered by the end of this article.

At first, we will learn about the idea of continuity with some simple and easy definitions in addition to some rigorous mathematical definitions of the continuity of a function on a domain or at a single point of its domain, after that, we will learn about the three conditions necessary to verify the continuity of a function at a point and the various properties of continuity. Next, we will proceed to learn about the discontinuity of a function alongside the different types of discontinuity, afterword we will introduce the theorem of intermediate values all this alongside taking a look at some illustrating examples in order to have a better understanding of the subject.

Well, one of the simplest ways to define a continuous function is that a continuous function is any function that has the characteristic that its graph can be drawn with a pen without needing to lift the pencil from the page, so for a function to be continuous on a domain or an interval the graph must be one single curved line and having one part not multiple on this interval i.e. there are no holes, breaks or gaps, of course, this definition isn’t a formal mathematical one but is used to simply explain the concept in an easy way.

For more formal, accurate, and a well mathematically put definition, we define the continuity of a function at a point as follow:

**Definition 1:**

Let be a function, let be its domain of definition, and let be a real number non isolated of ;

To say that the function continuous at the point , means that the limits of the function at the point is .

Meaning:

For better understanding let’s take a look at some examples:

**Example 1:**

We have the function defined on the interval as follow:

If we represent the function graphicly we get the following graph:

Notice the graph can be drawn without lifting the pen from the paper.

Therefore we can conclude that the function although being a composite function is continuous.

Let’s take a look at another example.

**Example 2:**

Let the function be defined as follow on the domain

Here is the graph:

If we take a look at the graph of the function we clearly notice that the graph can’t be drawn without lifting the pen of the paper.

Also, if from the first and second examples we can deduce that composite functions can be continuous on their domain of definition.

In addition, we notice that in order for a function to be continuous at a point, this point must be contained in the domain of the definition of the function. Meaning that in order to use the definition above we assume the existence of and .

This last point leads us to another definition of the continuity of a function at a point, which is the following:

**Definition 2:** Continuity at a point:

This means that in order for a function to be continuous at a point, the limits of when tends to a from both sides positive and negative side must be equal to .

From the previous examples, we can deduce the three conditions necessary in order for a function to be continuous, here they are:

**Definition 2:**

Suppose we have a function , we say that is continuous on a point , if and only if these conditions are verified:

- exists;
- exists;
- .

So, now that we defined what is continuity of a function at a given point of its domain of definition, we can now generalize the concept to define the notion of continuity of a function on an interval as follow:

*We say that the function is continuous on an interval if and only if is continuous on every point of the interval .*

__Continuity from the left:__

We describe a function as continuous from the left at the point if and only if:

- exists;
- exists;
- .

A graph of a left-continuous function

__Continuity from the right:__

We describe a function as continuous from the right at the point if and only if:

- exists;
- exists;
- .

A graph of a right-continuous function

Using this idea of continuity from the left and the right we can provide a definition for continuity on an interval.

**Definition:**

We say that a function is continuous on a closed interval , if and only if, these condition are verified:

- is continuous on ;
- is continuous from the right at the point ;
- And is continuous from the left at point .

Let’s take a look at an example:

Let’s determine if the function is continuous on the interval , with defined as:

First we verify that is continuous on :

We have

Therefor is continuous on .

Second, we verify that is continuous from the right at the point :

We have

Therefor is continuous from the right at the point .

Third, we verify that is continuous from the left at the point :

We have

Therefor is continuous from the left at the point .

So, the function is continuous on the closed interval

It is important and necessary to know the main properties of continuity because knowing them will make deducing the continuity of some complicated function easier and faster than trying without the use of these properties, here they are:

Here are some of the main properties of the continuity of functions:

- The addition: The addition of two functions and that are continuous on the interval (i.e. ) is a continuous function on .
- The subtraction: the subtraction of two functions and that are continuous on the interval (i.e. ) is a continuous function on .
- The scalar product: the product of a real number by a function that are continuous on the interval (i.e ) is a continuous function on .
- The product: the product of two functions and that are continuous on an interval (i.e. ) is a continuous function on .
- The division: we state two main cases:
- the division of two functions and that are continuous on an interval (i.e. ) and with different than 0 on , is a continuous function.
- The division of two and that are continuous on an interval and with for , is a discontinuous function.
- Composite function: the composition of two functions and that are continuous on an interval is a continuous function.
- The power: the power of a continuous function to where is appositive integer (i.e. ) is a continuous function.
- The root: the nth root of a continuous function where is a positive integer, is a continuous function.

These properties are easy to prove, so let’s take a look at a few proofs:

Let’s suppose that and are two continuous functions at a point .

We have (since is continuous at ),

And we have (since is continuous at ),

And we know that:

Therefore, the product function is continuous

The same thing if we take the the summation or the subtraction of the functions:

**Example 1:**

Let’s try to find out if the function is continuous:

We have that the function is the division of two functions and

Since these two functions are polynomials then we know that they are continuous on their domain of definition which is for both of them.

Now, all we need is to know if the denominator can have a null value;

We have the denominator can be null for two values, therefore the function is discontinuous (discontinuous on the points 1 and -1).

**Example 2: **

These functions are continuous:

; ; .

**Theorem:**

*Every Polynomial, Rational, Root, Exponential, Logarithmic, trigonometric, and inverse trigonometric function is a continuous function on its domain. *

A discontinuous function is a function that isn’t continuous.

In simple and easy put words we say that a function is discontinuous if the graph of the function can’t be drawn without lifting the pen, meaning that the graph has at least two separated parts.

A more rigorous definition is that a discontinuous function at a point is a function that the limit from the left at the point is different from the limit of that function to the right at the point .

Meaning:

Also, if a function isn’t defined at a point then it is discontinuous on every interval that contains the point .

Similarly to what we saw as conditions for a function to be continuous we can define conditions that are sufficient for a function to be discontinuous:

- If doesn’t exist at a point ;
- If doesn’t exist meaning: ;
- If , , and are different.

**Example 1:**

A known function that is discontinuous is the inverse function

By looking at the graph, we can clearly notice that it is composed of two separated (i.e. disconnected) parts meaning it can’t be drawn without lifting the pen.

The point for which the function is discontinuous is 0, and we have the domain of the inverse function is .

**Example 2:**

Let’s have a look at the graph of the function :

We have the domain of is

Notice that the graph has three separated parts, similarly to the domain of definition that is composed of the union of three intervals.

we can distinguish three types of discontinuity based on the form of the discontinuous graph in question.

We say that a discontinuous graph has a jump discontinuity if the graph is detached at a point and the limits of the function to the right and to the left of are not the same i.e.:

and therefore doesn’t exist.

Then is described as a jump discontinuity (or step discontinuity, or a discontinuity of the first kind).

Here is an example of a graph of a function that has a jump discontinuity:

We say that a discontinuous graph has a removable discontinuity at a point if the limits from both sides left and right at the exists and if they are equal i.e. but they are different than . in this case, the graph would look like a continuous graph except for a point detached point. Here is an example of a graph having a removable discontinuity:

We say that a discontinuous graph has an infinite discontinuity at a point if one or both limits from the left or the right at the point doesn’t exist, meaning one or both of these limits equal . This type of discontinuity is also called essential discontinuity or discontinuity of the second kind.

Here are some graphs that have infinite discontinuities:

In this section, we are going to take a look at some of the main theorems and results of continuous functions

**Definition:** *Absolute maximum and absolute minimum.*

We say that is an absolute maximum of the function on its domain , if and only if

Respectively, we say that is an absolute minimum of the function on its domain , if and only if

Now that we defined what is the maximum and the minimum absolute, we can introduce the following theorem:

**The theorem of extreme values:**

If the function is continuous and if the domain is compact, then the function has an absolute maximum and a minimum absolute on the interval .

**Theorem:** *Intermediate Value Theorem*

Suppose that the function is continuous and that , then there exists a real number which verifies that .

Respectively, is we have that is continuous and if , then there exists a real number which verifies that .

This theorem has a very important corollaries, here is one for example:

**Corollary:**

Suppose that is a continuous function; Let:

and

Thus for every there exists which verifies that .

Also, we can deduce from the theorem of intermediate value that if a function is continuous on the interval , and if and have different signs (one is positive and the other is negative) then we know for sure that the function has at least one root which is contained in the interval this result is also known as the “*Bolzano’s theorem*”.

**Theorem: ***Bolzano’s Theorem (1817)*

*If a continuous function defined on an interval is sometimes positive and sometimes negative, it must be 0 at some point.*

**Theorem:**

Suppose that is a strictly increasing and a continuous function on the interval , and let and , then is, one to one, ; and the inverse function defined on by

Is also a continuous function with the domain and to its image .

Well, here we finish our journey for this time in the world of calculus; we learned in this article about the concept of continuity of a function both at a point or on an interval, we saw the definition of continuity and the condition for with which we can determine if a function is continuous at a given point, not just that we also presented the main properties of continuity since they are essential and important to quickly and easily tell if a function is continuous or not; after that, we presented the meaning of discontinuity and the different types of discontinuity and how to detect them visually just by looking at the graph of the function in question, all of this alongside some examples and illustrations for a better understanding. We also learned some important theorems and results like the theorem of indeterminate value and some important results that we can extract from it. Well, it’s true that we have come to the end of this fun blog post but don’t worry at all, we still are just scratching the surface of Calculus and there is an infinity of things to learn, and we will ** continuously** keep doing that!!!!!

In the meantime, you can take a look at some articles about limits and infinity like Limit of a function: Introduction, Limits of a function: Operations and Properties, Limits of a function: Indeterminate Forms, or the ones about Infinity: Facts, mysteries, paradoxes, and beyond or check the ones about Probability: Introduction to Probability Theory, or Probability: Terminology and Evaluating Probabilities or maybe the one about Probability theory – Conditional Probability!!!!! !!!!!

There are also some articles about Set Theory like Set Theory: Introduction, or Set theory: Venn diagrams and Cardinality!!!!!

Also, if you want to learn more fun subjects, check the post about Functions and some of their properties, or the one about How to solve polynomial equations of first, second, and third degrees!!!!!

And don’t forget to join us on our Facebook page for any new articles and a lot more!!!!!

The post Continuity – Calculus appeared first on Math Academy.

]]>The post How to Improve Your ACT Math Score appeared first on Math Academy.

]]>Read the question carefully and underline the thing you’re asked to find.*Identify what you’re asked to find*.Often, a quick scan will reveal something that enables you to work more efficiently.*Scan the question-and-answer choices for patterns and shortcuts.*The best way to avoid mistakes is to work methodically.*Solve the question in an organized way.*

**How to overcome test anxiety:**Test-takers who struggle on the ACT math portion generally do so because they lack a clear understanding of the three distinct steps to solving the question. That lack of clarity creates test anxiety, which drives test-takers to rush through steps 1 and 2 or, worse, to skip those steps entirely. It’s vital that you’re aware of the role that anxiety plays in driving some test-takers to abandon their training on test day. If you allow anxiety to determine your approach or set your pace, your score will be affected.**How to develop good habits:**As you begin prepping for the math portion, we strongly recommend that you commit to working slowly and deliberately through each step. As you do, pay close attention to the feeling you experience each time you approach a question. Notice whether you feel an urge to rush through steps 1 and 2 or to skip them entirely. It’s critical that you learn to recognize these impulses and remain calm instead of charging ahead without a plan. Be sure to develop these skills before you shift your attention to speeding up your pace.**How to improve your pacing:**Focusing on speed before you’ve developed good habits can make it more challenging to master the skills you need to become a more efficient test-taker. This is an important distinction: efficiency and speed are not the same thing. In ACT prep, improving efficiency increases test scores. Obsessing about speed increases anxiety and mistakes, both of which lower test scores. So, as you begin this section, commit to slowing down, focusing on steps 1 and 2, and remaining calm as you solve each question.**How to optimize your score:**Over time, as your efficiency improves, you will naturally pick up speed — and because that speed is the result of efficiency, it won’t come at the cost of increased mistakes. On the contrary, the number of mistakes you make will decline because you’ll know what to solve and how to avoid performing unnecessary work. Eventually, once*identifying, scanning,*and*solving*have become habits, you can turn your attention to optimizing your score by picking up speed on steps 1 and 2 without rushing to step 3 and committing errors.**How your attitude can predict your score improvement:**Especially in the math section, the biggest difference between students who improve and students who don’t is their attitudes. Students who fail to improve react to a wrong answer by getting frustrated or by dismissing the wrong answer as a “silly mistake.” In either case, the reaction prevents the opportunity to learn from that mistake. In contrast, students who improve their scores tend to react to wrong answers with an eagerness to discover their mistakes and figure out how to avoid making the same mistake in the future

If you’ve scored 30 or above (95th percentile) on a timed, authentic ACT practice test, then you may want to use *Approach 2*. That said, you should definitely read and consider *Approach 1 *before you make your decision. For everyone else (95% of all test-takers), you’ll want to use *Approach 1*. But before we get to that, you need to understand a few very important facts about question difficulty, trends in performance, and raw/scaled scores:

- The difficulty of ACT math questions increases throughout the section, from questions 1-60.
- Questions 1-20 are mostly easy with a few medium questions mixed in — especially near the end. • Questions 21-40 are mostly medium with a few easy questions mixed in near the beginning and a few harder questions mixed in after question 30.
- Questions 41-60 are most difficult with a few medium questions mixed in near the beginning.
- Questions 55-60 are all very difficult.
- There is rarely more than 1 hard question before 30 or 1 easy question after 30.

The good news for most students is that you don’t have to solve every question correctly. You don’t even need to attempt every question. In fact, you probably shouldn’t attempt every question. To be clear, you should never leave a question blank, but that doesn’t mean you have to waste time trying to solve a question that’s beyond your understanding. As you consider the following stats, keep in mind that random guessing has a one-in-five chance, which you’d expect to result in correct answers 20% of the time.

- Students scoring below 23 correctly answer < 50% of questions after #30 and < 30% after #40.
- Students scoring 23-29 correctly answer < 50% of questions after #50 and < 25% after #55.
- Students scoring above 30 correctly answer < 50% of questions after question #58.

As you can see, most students are actually better off skipping a significant number of ACT math questions. We recognize that some students won’t like the idea of planning ahead of time to guess on 5, 10, or even 20 questions, but the stats are clear. All students, even top-performing test-takers, hit a wall at some point during the ACT math section. The key is to identify where your “wall” shows up on the test, so you can plan ahead and make the best use of your time.

Sample Raw Scores Compared to Scaled Scores: The raw score (number of correct answers) required to earn a good score on the ACT Math section is not as high as you might expect. Again, this all points to the fact that most students shouldn’t plan to waste time on the most difficult problems on the test. Instead, that time should be put to better use by making sure you don’t commit any errors on the easier questions that you are more likely to be able to solve.

This approach is best for the vast majority of test-takers. The idea is to spend more time on the easy and medium questions, rather than wasting that time struggling to understand harder questions. For these students, there’s no point in rushing through the easy and medium-difficulty questions to get to the hard questions that they don’t know how to solve.

The goal is to find the point after which you miss more than 50% of all the remaining questions. Sometimes, that’s obvious, but most of the time you’ll need to make a few basic calculations.

The easiest way to find your “wall” is to work backward from the end. Start by calculating the percentage correct for 60-51. Less than 50%? Try 60-41. Greater than 50% correct? That’s great! Try 60-45.

Let’s say you hit the wall after question 45. That’s not bad. If you can answer 40/45 before you hit the wall and then guess right on 3/15 (20%) of the remaining 15 questions, you’ll end up with 43 correct answers, which converts to a score of 27, putting you in the 88th percentile — and remember, that’s if you guess blindly on a quarter of the math section!

In the above example, we recommend skipping the last 15 questions, which leaves you with an extra 15 minutes to spend however you want on the first 45 questions. If you allocate that time evenly among all 45 questions, you’ll have 1.25 minutes per question (+25%). However, if you maintain your original pace of 1 minute per question for the easy questions (1-20), then you can use all 15 minutes to solve the medium and hard questions (21-45). This plan gives you a whopping 1.6 minutes (+60%) per question for 21-45 — and all it cost you was 15 minutes of wasted time!

A pacing strategy only works when it is simple enough to follow without becoming a distraction. So, let’s keep it simple. Through the first 20 questions, make sure that the time elapsed (in minutes) stays below the question number you’re solving. If you keep that pace, you’ll complete question 20 with 25 questions and 40 minutes remaining. That’s a ratio of 5 to 8, meaning you need to solve 5 questions every 8 minutes. You could check the time after every fifth question, but we recommend checkpoints after question 30 (36-min) and question 40 (52-min). From there on, just work as quickly as you can, and be sure to save a few seconds to bubble in guesses for questions 46-60!

If you are already scoring above 30 on authentic, timed ACT math tests, then first of all, congratulations!

Now, let’s get to work. Once you’re above 30, the best approach to maximizing your score is basically the opposite of *Approach 1. *Instead of freeing up time for easy and medium questions, *Approach 2 *is all about getting to the difficult questions with as much time as possible remaining.

Unlike *Approach 1*, however, you can’t just skip a bunch of easy and medium questions to free up time for the hard ones. Instead, you must answer every easy and medium question as efficiently as possible in order to *BANK *your time for the harder questions near the end of the section. Note: you should be answering more than 90% of the first 40 questions correctly. If that’s not the case, you should consider using *Approach 1 *and/or reviewing the steps mentioned.

**1. Review your initial diagnostic test results.**

Follow the steps described in *Approach 1 *to find where you hit a wall. For most students scoring above 30, that point comes somewhere around question 55. The higher your initial score, the later the wall. Whatever the case, it may still be advantageous for you to skip/ignore the final 2 or 3 questions and apply that time to make sure you answer the rest of the questions from 51-60 correctly. That’s your decision to make with your tutor.

**2. Efficiency is everything**

Focusing on “speed” is the wrong approach. Any time that you save will most likely come at the cost of increased errors. Of course, that defeats the purpose of speeding up, since the only reason to go faster is to allow yourself more time to earn points on harder questions! If you focus on *efficiency*, however, you’ll ultimately increase your speed without making errors.

**3. Execute your optimal pacing strategy**

Top scores are won or lost early on the test when you’re banking time for those last 10 questions and before that when you’re deciding whether to attempt all 60 questions. Your objective is to develop *a sense *of exactly how quickly you are able to solve math questions without making errors.

Unlike *Approach 1*, there is no formula you can use to determine your maximum safe speed. During the test, you will need to be able to *feel your pace *as you go. You can still set a few checkpoints to make sure you don’t get too far ahead or behind, but you will still need to be able to *feel your pace*, so you can stay balanced between too speedy and too cautious*. *Ultimately, there’s no point to pushing yourself to ace the last 10 questions, if doing so causes you to commit mistakes on easy and medium questions.

With that in mind, pay careful attention to pacing as you complete your workbook lessons, quizzes, and practice tests, so you can get more comfortable working at your maximum safe speed.

Below, we explain what to look for as you scan question-and-answer choices. Learning to spot the patterns and shortcuts on this list can really improve your score. Let’s take a look!

- From A to E or F to K, the answer choices are arranged in ascending order (i.e., least to greatest)

You may have already known this, but you probably weren’t aware of all the ways this simple pattern can help you solve difficult questions more efficiently. For example, when you’re solving backward by plugging in the answer choices, you can work methodically to minimize the number of answer choices you have to test.

- Pay attention to differences and
__similarities__among answer choices

Each of the answer choices will be different, obviously. That said, a number of them may be similar in important ways. In only 2-3 seconds, you may notice a pattern that shows you how to approach the problem more efficiently, saving you an entire minute. Equally important, you may notice that two or more answer choices are very similar. In this case, make a mental note to be very careful when you match your solution to the answer choices after solving the problem.

- Be sceptical of “eye-catching” answers that reuse numbers from the question

The test-makers know that some students will be attracted — consciously or subconsciously — to answer choices that *feel related *to the question. Because test-makers know that eye-catching answer choices appeal to lazy test-takers, these answer choices are often incorrect.

- Pay attention to how the answer choices relate to each other, especially:

Multiples or factors (e.g., 11, 22, 44, 110). For example, if you notice that all of the answers are multiples of 11, then that is a clue as to how you should approach the question. So if you end up with 19, that’s a signal that you had a bad approach (as opposed to a good approach and a miscalculation).Pairs of positive/negative numbers (g., 4 and –4 or 1/7 and –1/7). When you see answer choices that are identical except for their +/– sign, this is a warning that you’ll need to be careful with your signs and when multiplying by negatives.Pairs of “flipped” answer choices (e.g., 2/9 and 9/2 or

*m/n*and*n/m*). Often, one of the flipped answer choices is correct and the other is there to catch test-takers who made an error. Furthermore, some percentage of test-takers incorrectly match their correct answer to the answer choices and bubble in the letter corresponding to the flipped version.Pairs of complementary angles (e.g., 20º and 70º), supplementary angles (e.g., 46º and 134º), or explementary angles

*(e.g., 330º and 30º).*This often indicates that one of the two is the correct answer, while the other one is just there to trap students who solved for the wrong part of the figure.Pairs of percentages that add to 100% (e.g. 60% and 40%). These trap test-takers who forget what they’re supposed to find and solve for the discount percentage or the percentage of the original price represented by the new price. The same goes for non-percentage answer choices (e.g., discount, the new price, the new price + sales tax).

- Answer choices that form a series (g., 15, 18, 21, 24). Consider a question that asks you to find the perimeter of a regular hexagon with a side length of 3. To find the answer, you would multiply 6 (the number of sides) by 3 (length of each side): 6 x 3 = 18. That’s what
*you would do.*But the test-makers know that some students don’t know how many sides a hexagon has. They might end up with 15 or 21. Other students may choose to punch in 3+3+3+3+3+3 = 18. That method produces the correct answer, but it’s slower – and way more dangerous as you will see! That’s because every calculator keystroke represents another chance to make an error. The more keystrokes, the more chances to mess up the calculation. For example, some students will punch in the wrong number of 3+3s, ending up with 3+3+3+3+3 = 15 or 3+3+3+3+3+3+3 = 21. In either case, these test-takers will almost always find their wrong answers conveniently waiting for them.

Remember that the answer choices are organized from least to greatest. So, if the correct answer, 18, was choice A, then there is nowhere to put the trap answer 15. As a result, test-takers who mistakenly ended up with 15 would not find their answer, which means they’d get a second chance to solve the problem. Following the same reasoning, the correct answer is not likely to be choice E because that would knock the trap answer 21 off the list of answers, giving a second chance to all the test-takers who ended up with 21.

The takeaway is that when you see answers in a series, you should *lean toward *B, C, or D because they’re slightly more likely to be the correct answer. Obviously, you should solve the problem if you can, but if you’re stumped, choose B, C or D. Of course, that doesn’t mean that it will never be A or E, but if you have to guess, you should choose B, C, or D.

- Equations translated from a word problem
If you’re not already a pro at it, you should definitely learn to translate word problems into equations. While translating is the most reliable way to solve a word problem, you may find that it’s more efficient to methodically check how a single variable from the word problem shows up in each answer choice. This is especially helpful for eliminating wrong answers on word problems that deal with variable cost and fixed cost. In these cases, you can usually eliminate one or even two answers that mishandle the fixed cost. Any answer choice that multiplies the fixed cost by any other number is wrong.*:*

- Don’t just retrace your footsteps If you do complete the section with time remaining, you should review your answers. Start with any answers you weren’t certain of or that you guessed between two remaining answer choices.

There are two ways to review an answer: retracing your steps or solving the problem a new way. Between the two, solving the problem again is much more likely to help you uncover an error, especially if you solve it using a different method. For example, if you solved a simultaneous equation problem using substitution the first time, you should use elimination on your second pass. Reaching the same answer a different way is more reliable than solving it the same way and possibly repeating the same error you made the first time.

- When you’re short on time If you only have a few minutes left, you need to make quick decisions about which problems are most likely the best use of your time. Remember that difficulty is not necessarily a measure of the time it takes to solve the problem. For example, an easy question may not be quick to solve and a difficult question may only require a very simple calculation once you figure out what you’re asked to find. Regardless of a question’s difficulty, you can generally assume that the more text in the question and the more complicated the answer choices, the more time-consuming that question will be to solve. So, if you’re down to the wire, scan the page and target the question with the least text and least complicated answer choices.

The post How to Improve Your ACT Math Score appeared first on Math Academy.

]]>The post Probability theory – Conditional Probability appeared first on Math Academy.

]]>**Table of content:**

- Introduction
- Types of events
- Conditional probability
- The formula of conditional probability
- Properties of conditional probability
- Bayes’ theorem
- Application of conditional probability
- Conclusion

We learned previously that calculating the probability of an event is simple, knowing the sample space and the favorable outcomes. It is straightforward, all it takes dividing the number of favorable outcomes by the total number of outcomes. But what if the event we are evaluating its probability is related to another event, or what if we receive new information from the previous trials, does this information change or affect the probability of the event we are investigating?! Does all information affect the probability of the event we are currently investigating?! And how to evaluate an event if it is dependent on another one?! We will answer all these questions and more in this article.

First, we will take a look at the different types of events and the various possible types of relations between two events, after that we will proceed to introducing the concept of conditional probability and its meaning, we will introduce the formula of conditional probability alongside an explanation to make it trivial and easy to grasp and of course, we will take a look at some illustrating examples to strengthen and test our understanding of the conditional probability, afterword we will present the main properties of conditional probability, also we will introduce the Bayes’ theorem, and finally we will learn about the usage and applications of conditional probability in the different sciences and fields.

We call a “sure event” every event that its probability is equal to 1, meaning, and as the name suggests, that we are sure of the occurrence of this event. On the other hand, we call an “impossible event” every event that its probability is equal to 0, in other terms, and as the name suggests as well, we are certain that the event will not occur.

An example of a sure event is the event that the sum of rolling of two dice is less or equal to 12.

An example of an impossible event is the event of getting a number greater than 6 on a roll of a die.

We call a “simple event” all event that is a one and only one element of the elements of the sample space.

For example, the event of getting the number 5 from a roll of a fair die is a simple event since 5 is one element of the sample space , but if we take the event of getting a prime number as a result of a roll of a die, this event is not a simple event since the event E is composed of multiple elements of the sample space , so if we write the event E as a set we get: meaning that it has three elements of the sample space and not just one.

This is the opposite case of the simple space, we call a “compound event” every event that has more than one element of the sample space, in other terms a compound event is an event that contains multiple outcomes of the possible outcomes of the sample space.

For example, the event of getting an odd number after the rolling of two fair dice is a compound event since it contains six elements of the sample space, so if write the sample space we get and we have the set of the event can be written as , we can clearly see that the event is a compound one.

But, if we take the event of getting a “Tail” after a fair coin toss, this event is not a compound one since it has only one element of the sample space (and therefore it is a simple event).

We describe events as “independent” in the case where the happening of any event doesn’t affect the chances of occurrence of any other event, and we call them “independent events”. In other terms, for two events to be called independent one must have no influence on the chances of occurrence of the other.

For example, if we take the case of rolling a fair die, we know that every roll is completely unaffected by the result of the previous roll, so the chances of occurrence of the number 2 for instance, is the same no matter the number we got from the last roll.

Another example is the toss of a fair coin, it is clear that every toss is completely separate from the next one, and we know that every toss is not affected whatsoever by the result of the previous toss, so if we take for instance the chances of getting “Tail” after getting fifty-three times in a row the result “Head”, the chances are still the same, and we still have .

Another one is getting a free lunch at your favorite restaurant and the weather becoming rainy; winning the lottery and the occurrence of a solar eclipse that day; or raining and running out of milk.

We describe events as “dependent” in the case where the probability of occurrence of any event is affected by the happening of another event and we call them “dependent events”. In other terms, for two events to be called dependent one must have an influence on the chances of occurrence of the other.

So, one good example of dependent events is the following: if we consider having an urn containing 4 white balls, three blue ones, and 7 purple ones, if we take out a ball without looking and without returning it back to the urn, and we consider and we consider the event A of taking out a purple ball, well, at the first take out we have the probability of this event as usual which is the number of favorable outcomes divides by the total number of the possible outcomes, so we get:

But if we consider a second take out, ** knowing that we took a purple ball in the first time**, the probability is affected, it is no more because when we took out one ball, we didn’t put it back in the urn and therefore we have now 6 purple instead of 7 and we have a total of 13 balls instead of 14.

Another example of two independent events is the following: suppose we have a giveaway and we have 100 participants, the giveaway has to parts: at the beginning, 50 people will be chosen randomly to win a headphone for each, and after that 3 lucky people from the winners of the headphones will win an iPhone 13 Pro. Well, notice that the event of winning an iPhone 13 pro is dependent on the event of winning the headphones since in order to be considered for the iPhone giveaway you need to be already a winner of the headphone. In other terms, the chance of winning an iPhone 13 pro is affected by the chance of winning a headphone.

We describe two events as “mutually exclusive”, if the occurrence of one event implies the non-occurrence of the other one, meaning that if one event happens, we know for sure that the other one didn’t, and vise-versa. In other terms, the occurrence of one event excludes the occurrence of the other and hence they are called “mutually exclusive events”.

For example, if we have the experiment of rolling of a fair die and if we consider the two events and as follow:

: the event of getting an odd number.

: the event of getting an even number.

Also, we write sample space and the two events and in the form of sets, we have:

, and .

Then we know that the two events and are mutually exclusive since that we can’t have a number as a result of a roll of a die that is at the same time odd and even. And also, we know that the two events are mutually exclusive by looking at the elements of each event and noticing that there is no common element and thus they can’t occur at the same time (The intersection of and is the empty set).

Another example is the toss of a coin, we have one of two events: “Head” or “Tail” and the occurrence of one of them implies the non-occurrence of the other since we can’t have the two results together.

Some other examples are: winning and losing the lottery, driving forward and backward, …etc.

Suppose we have a sample space and an event , we call the “complementary event” of , the event that contains all the elements of the sample space other than the ones contained in . In other terms, the complementary event of is the set of the remaining elements of S if we remove the ones that are in .

We can write .

Notice that the two events and are mutually exclusive.

For example, if we consider the event of having a number less or equal than 4 as a result of a rolling of a fair die. We have the sample space : , and we have the event E1 can be written as , thus we have the complementary event of as follow:

; .

Another example is having the experiment of rolling a fair die and considering the event of getting an even number, we know that the elements that are in the sample space and not even can’t be anything other than odd numbers, and therefore the complementary event of getting an even number as a result of the roll is the event of getting an odd number.

After taking a look at the different types of events let us now learn what is conditional probability.

Conditional probability is defined as the probability of occurrence of an event knowing that another event has already happened, different from performing the experiment for the first time, in the case of conditional probability we have additional information that is related and can affect the evaluation of the chances of an event occurring. Knowing this new information gives us a new insight and pushes us to calculate the probability in question while taking the new information into consideration. Bear in mind that conditional probability doesn’t imply that there is always a causal relation in the midst of the two events, there are several cases depending on the types of events and the relation between them. Conditional probability opens the question: as we get new and additional information does this allow gives us the possibility and the potential of reevaluating the probabilities we actually have and get a better estimation of the chances of occurrence of the coming events?!

Thus, the goal of conditional probability is finding the probability of an event ** given that** or

If we have two events and the probability of occurrence of , provided the information that has already happened is noted by the notation: and we read: the probability of the event given the event .

Let’s take a quick example: If we have an urn containing four balls: white, blue, orange, and purple, in this experiment we draw without returning the drawn ball.

And we want to evaluate the probability of drawing the orange ball after already drawing the purple one.

First, we note the following events:

: The event of taking out the purple ball.

: The event of taking out the orange ball.

Now, and assuming that the balls have the same chance of getting taken out, we have that the probability of drawing the purple ball is .

But assuming that the event of picking the purple ball occurs, now we have three remaining balls which imply a change of the probability of picking one of them it is no more , instead it is now .

So the probability of drawing the orange ball after already drawing the purple one is .

For more understanding let’s take a look at this example: Suppose that a fair die has been rolled and we want to evaluate the probability of getting a 2. Typically and since the die is fair the probability is . But if we are given the information that the resulting number from the roll is an even number! Now we do not still consider the sample space as if it still has all six possible outcomes, but instead, we now have only three possibilities which are 2,4, and 6, and thus the probability becomes equal to instead of .

Now suppose we got different information, we got informed that the resulting number of the roll is an odd number! In this case, we know that the chances of getting a 2 are impossible meaning .

This demonstrates how the new information can influence the probability and help to get a better and more accurate estimation of the occurrence of the event in question.

The conditional probability has a formula that helps us to evaluate the wanted probability easily, let’s take a look at it:

Suppose that we have two events and , the conditional probability is given as follow:

Where:

: the probability of the event given the event .

: the probability of the intersection of and .

: the probability of .

The formula of the conditional probability is derived from the multiplication rule (also called chain rule), which, if we are using a tree diagram, means that in order to evaluate the probability of an event at any point of the tree diagram by multiplying the probabilities of the branches leading to that point. This also shows how useful and important the tree diagram is for conditional probability problems and for the probability theory in general. The multiplication rule is given by the following formula:

And therefore, by dividing the formula by we derive the formula for the conditional probability .

Here is a tree diagram illustrating the multiplication rule:

So, an easy visual way to calculate conditional probability is by using the tree diagrams.

From the definition we saw of independent events, we know that one event doesn’t have any effect on the probability of occurrence of the other, and therefore the conditional probability of the events A and B and knowing that the two events are independent is given as follow:

As we learned earlier, mutually exclusive events can’t happen at the same time, and that the occurrence of one implies the non-occurrence of the other, and thus we have the conditional probability of two mutually exclusive events and is given as follow:

By using the multiplication rule we can evaluate the probability of various cases, and thus by using the multiplication rule we can derive what is called the Law of total probability.

Let’s suppose that we have a sample space and that it is divided into three disjoint events noted ,, and ; therefore we have the following result for any given event :

And by replacing the terms on the right using the multiplication rule we get the expression for the law of total probability:

Here are a few properties of conditional probabilities:

Suppose we have three events , and , and we have , we have the following properties:

- if then

The British mathematician Thomas Bayes from the 18^{th} century developed a mathematical formula that helps us to determine the conditional probability (also known as Bayes’ law or Bayes’ rule), this formula is given as follow:

Suppose we have two events and , and , then:

This theorem is fundamental, important, and of great use in order to estimate the probability of an event based on new knowledge and information that may have a relation with the event in question. This helps us update the probabilities and estimations and get more accurate and updated predictions and revise the present ones knowing the new information.

Conditional probability is one of the most important and fundamental concepts of the probability theory, and in many other fields and sciences since it deals with the idea of the existence of a relationship between the events that may cause a change or make a difference in the estimation of the probability of the events in question, and this idea of the existence of a relation between things is what is most common in sciences and in the real-life in general. Also, the Bayes’ theorem is widely used in various fields like statistics and finance, banking, sports, health care, and genetics to name just a few. And with the growing businesses dealing with huge data and information the importance of probability theory and its various branches is more and more clear and apparent.

Well, here we come to the end of this article, and after taking this fun journey with conditional probability and the different properties, and learning about the importance and usage of conditional probability in real life, it is safe to say that we are extremely excited to learn more about probability theory and wander through the world of probabilities!!! Don’t worry at all, more fun articles are coming soon!!!

In the meantime, you can take a look at some articles about limits and infinity like Limit of a function: Introduction, Limits of a function: Operations and Properties, Limits of a function: Indeterminate Forms, or the ones about Infinity: Facts, mysteries, paradoxes, and beyond or check the ones about Probability: Introduction to Probability Theory, or Probability: Terminology and Evaluating Probabilities !!!!!

There are also some articles about Set Theory like Set Theory: Introduction, or Set theory: Venn diagrams and Cardinality!!!!!

Also, if you want to learn more fun subjects, check the post about Functions and some of their properties, or the one about How to solve polynomial equations of first, second, and third degrees!!!!!

And don’t forget to join us on our Facebook page for any new articles and a lot more!!!!!

The post Probability theory – Conditional Probability appeared first on Math Academy.

]]>The post Probability: Terminology and Evaluating Probabilities appeared first on Math Academy.

]]>**Table of content:**

- Introduction
- Terminology
- Finding the probability of an event
- Conclusion

After introducing the probability theory in the previous article: Probability: Introduction to Probability Theory, we continue our journey by diving more and learning the basics, in this article, we will continue the introduction and learn some terminology alongside the basic method of evaluating and calculating the probability of a given event.

We mean by experiment action of performing or conducting a test, an evaluation, or an investigation, and we get a result by the end of the experiment. an experiment may have one or many possible results, also, based on the number of possible results we can define two types of experiments: the deterministic experiments and the probabilistic ones.

We call a determinist experiment every experiment or a trial that we can predict with certainty the outcome experiment in advance, meaning throughout the experiment, there is no part ruled by chance or chaos and thus we can determine with certainty the outcome of the experiment even before it happens if we know its inputs or variables.

**Examples of deterministic experiments:**

- The addition of two numbers and ; for instance, the addition of 8 and 21 is certainly 29, there is no maybe in there.
- The multiplication of two numbers;
- The experiment of decomposition of water, we know for sure that we will have in result oxygen and hydrogen.
- Science experiments of established laws like conducting an experiment to test Newton’s laws Motion or gravity …etc.

We call random experiment (or probabilistic experiment) every experiment or a trial where we can’t foretell the outcome of the experiment beforehand, in other terms, throughout the experiment, there is one or many parts of it ruled by chance and randomness and therefore we can’t predict with certainty the result of the experiments before the experiment is done.

**Examples of random experiments:**

- Rolling a die, we know the possible results are the numbers from 1 to 6 but we can’t predict with certainty the result.
- Tossing a coin, we know that there are two possible results Head or tails, but we can’t determine beforehand the outcome of a toss.
- Selecting a numbered ball from an urn containing balls numbered from 1 to 100, of course without looking we can’t foretell the number of the ball we are going to take out.
- Taking out a card from a well-shuffled deck of cards.
- The winner of a car race.
- The winner or winners of a lottery.
- The score of a basketball game.

As we already know by now, probability theory is the study of the chance of occurrence of events, so we will be interested in studying the random experiment and trying to determine the different probabilities.

We call a trial the act of performing the experiment, therefore when we conduct a random experiment several times every one of them is called a trial.

We call an outcome every possible result of the experiment.

**For example**: if the experiment is to roll a die then the possible outcomes are: 1, 2, 3, 4, 5, or 6.

We call the sample space, the set containing all possible outcomes.

**For example:** for the experiment of rolling a die, the sample space is .

An event is a set of outcomes of the experiment, or in other terms, an event is a subset of the sample space set, in other terms we can say that an event is a collection of outcomes that have a common property. if the result of the experiment is contained in we say that the event has occurred.

For example: let consider the experiment of rolling a die, and we are interested in the outcomes where the number we get is even; we have the sample space is and if we denote the event of having an even number by we have . in this example, the event is a collection of outcomes that have the same property which is being even.

Note that, if we consider two events and then their union and their intersection are also events.

Let’s suppose we have an experiment with its sample space, to determine the probability of a given event , we need to extract the important and essential information about the experiment, and then by using the probability axioms we proceed to evaluate the wanted probability.

Let’s take a look at an example:

If we take the experiment or roll a fair die, and we want to determine the probability of the event where we get an odd number, the information given is:

- Fair die which leans that all the outcomes (numbers from 1-6) have the same chance of occurrence, therefore .
- The event is getting an odd number, so we can note .

Now since we have all the outcomes are equiprobable (meaning equally likely) we can to use the fundamental rule of evaluating probability:

In this case, we have the favorable outcomes are the outcomes that result in the occurrence of the event which means the favorable outcomes are: 1, 3, and 5 so we have three outcomes from the total number of outcomes which is 6.

Therefore, we have:

Note that the probability of the event where is

**Example 2:** Let’s determine the probability of taking out an ace from a well-shuffled deck of cards.

So if we analyze the information we have:

- The deck of cards contains 52 cards.
- There are 4 aces.
- The deck is well-shuffled therefore the chance of drawing any card is the same as any other card.

So by using the formula of probability we have:

- The number of favorable outcomes is 4.
- The number of the total outcomes is 52.

Therefore the probability of the drawing of an ace is i.e.

**Example 3:** Let’s consider the event of getting at least one tail in three tosses of a fair coin.

The information we have are:

- A fair coin thus the chance of having Head or Tail is the same.
- The experiment consists of three tosses therefore the sample space is .
- The event we want to determine its probability is getting at least one Tail, meaning .

So by using the formula of probability we get:

- The number of favorable outcomes is 7 (i.e. cardinal of ).
- The number of the total outcomes is 8 (i.e. cardinal of ).

Therefore the probability of getting at least one tail after three tosses of a fair coin is .

In this article, we learned the basic terminology of the probability theory; we also learned how to evaluate probabilities of given events by extracting the important and essential information and using the probability axioms to determine the probability of the event in question. keep in mind this is still an introduction to probability theory and we have much more to learn, and yes!!! there will be more articles on this subject!!!!!

In the meantime, you can take a look at some articles about limits and infinity like Limit of a function: Introduction, Limits of a function: Operations and Properties, Limits of a function: Indeterminate Forms, or the ones about Infinity: Facts, mysteries, paradoxes, and beyond or check the one about Probability: Introduction to Probability Theory!!!!!

Also, if you want to learn more fun subjects, check the post about Functions and some of their properties, or the one about How to solve polynomial equations of first, second, and third degrees!!!!!

And don’t forget to join us on our Facebook page for any new articles and a lot more!!!!!

The post Probability: Terminology and Evaluating Probabilities appeared first on Math Academy.

]]>The post Set theory: Venn diagrams and Cardinality appeared first on Math Academy.

]]>**Table of content:**

-Introduction

-Venn diagrams

-Cardinality

-Conclusion

After learning about the relations between sets and the operations on sets and their properties we will learn in this second article the representation of sets with the Van diagrams, we will also introduce the concept of cardinality and we’ll have a look at the importance and usage of set theory.

Quote: “A set is a Many that allows itself to be thought of as a One.” -Georg Cantor

A widely used way to visually represent sets and the relations between them is using Venn diagrams, this way of representing is named after the English mathematician and logician John Venn, who lived in the 18th century.

this way of representing the sets and the different relations between them is simple and easy to apply, we can summarize it as follow:

1-We draw a circle to represent a set, an ordinary one meaning not a universal one, inside the circle, we find the objects or elements of the set. The different circles representing the sets can overlap which means that these sets intersect.

Note that, a universal set is a set that contains all the elements that may be taken under consideration, in set theory to define the domain in which the operations on sets are done is to define the universal set. it is usually denoted .

**Definition:** We call a universal set the set of all the possible objects.

2-We draw a rectangle to represent the universal set, inside of it we find all subsets and elements.

Here are some illustrations of the main operations on sets:

- The union of and :
- The intersection of and :

- The difference of and :

- The Symmetric difference of and :

A special and simple aspect of sets that mathematicians are always interested in, is the total number of distinct elements of a set whether it is a finite set or infinite. This property is very important since it allows us to know the size of a set and compare the different sets among many other things.

At the first look, this idea of cardinality is a very simple one we just need to count the elements of a given set to determine its cardinality, and that is totally true when we are dealing with finite sets, but the idea of cardinality isn’t that clear and simple to grasp when it comes to infinite sets and comparing the size of infinite sets!

The cardinality of a set is noted and sometimes in some references, it is noted by .

Note that the cardinality of the empty set is 0:

Note also that we count only the distinct elements so for example, .

First let’s take a look at some illustrating examples:

**Example 1:** Let the universal set be: , and

Then we have the cardinality of the universal set is (meaning that has 9 distinct elements).

And the cardinality of is (meaning that has 6 distinct elements).

It makes sense that the cardinality of the universal set is greater (or equal) than any other one since any other one is a subset of the universal set.

**Example 2:** Let be defined as follow:

We have the cardinality of :

(note that we can write in a different form such as: ).

In case we have two sets and with the same cardinality we note and we say that and have the same size. Be careful between and ; means that and have exactly the same elements, for example, if and and then we have . For the expression it means the number of elements of is equal to the number of elements of but it doesn’t necessarily mean the same elements, for example, if and we have but and have different elements.

Therefore, implies that but not the other way.

For the case of infinite sets, cardinality has some interesting properties, for example, we can have two infinite sets and and yet the number of elements of is bigger than the number of elements of i.e., ; this is a fundamental result of set theory for the study of limits and their properties. for more understanding check out our article about Infinity: Facts, mysteries, paradoxes, and beyond.

we can easily conclude that:

- Every subset of a finite set is a finite
- Every uperset of an infinite set is an infinite

let and be two sets, we have the following properties:

- The sum of the cardinality of and the cardinality of is equal to the cardinality of the union of and :

- The product of the cardinality of by the cardinality of is equal to the cardinality of the cartesian production of and :

-Assuming the axiom of choice, we have the following result:

The cardinality of the union of and is equal to the cardinality of the cartesian product of and and it is equal to the maximum between the cardinality of and the cardinality of :

.

For finite sets , and we have the following properties: The Inclusion-Exclusion principle:

- For the case of two finite sets:

- For the case of three finite sets:

Let’s take a look at an example for a better understanding of the Inclusion-Exclusion principle:

Suppose that at a birthday party we have the gests meet the following description:

- We have 12 people wearing blue shirts and 9 people wearing green shirts;
- 5 guests have blue shirts and black jeans;
- 4 guests have black jeans and green shirts;
- The total number of guests with a blue shirt or a green shirt or black jeans is 25.

And we want to determine the number of guests that have black jeans!

To answer this question, we take the following steps:

Let be the set of guests with blue shirts

Let be the set of guests with green shirts

Let be the set of guests with black jeans

Now using the notation, we can write the information given as follow:

; ; ; ; .

we can safely assume that and are disjoint, meaning since guests are either wearing a blue shirt or a green one not both at the same time.

Now using the principle of Inclusion-Exclusion we get:

Therefore

So, the number of guests wearing black jeans is 13 (5 of them are wearing blue shirts and 4 of them are wearing green shirts and the rest (4 guests) are wearing neither a blue shirt nor a green one).

-The set of natural numbers and the set of odd numbers have equal cardinalities i.e. , and yet is a subset of ()!

-The set of natural numbers and the set of even numbers have equal cardinalities i.e. , and yet is a subset of ()!

-The set of natural numbers and the set of integers have equal cardinalities i.e. , and yet is a subset of ()!

-The set of real numbers and the interval of real numbers have the same cardinality, i.e. , and yet is a subset of ()!

-Any two open finite intervals of real numbers have equal cardinalities, i.e. . For example, .

In this article we learned about the visual representation using Venn diagrams; we also learned about the concept of cardinality and some of its properties in the case of finite or infinite sets, and how it allows us by counting the number of distinct elements of sets, to compare their sizes either they are finite or infinite. This article shows a part of the importance of set theory and how it is fundamental for studying limits and comparing them, it is also fundamental for Algebra, logic, and probability theory among many other branches of mathematics and other sciences. Keep in mind that this is merely scratching the surface of set theory and that the set of things to learn about set theory has an infinite cardinality!!!!!

In the meantime, you can take a look at some articles about limits and infinity like Limit of a function: Introduction, Limits of a function: Operations and Properties, Limits of a function: Indeterminate Forms, or the ones about Infinity: Facts, mysteries, paradoxes, and beyond or check the one about Probability: Introduction to Probability Theory!!!!!

And don’t forget to join us on our Facebook page for any new articles and a lot more!!!!!

The post Set theory: Venn diagrams and Cardinality appeared first on Math Academy.

]]>The post Set Theory: Introduction appeared first on Math Academy.

]]>Set theory is the mathematical branch that studies the sets and their properties, the operations on sets, the cardinality among many other sides sets. The beginning of the modern set theory was around 1870. Set theory is a fundamental branch for the entire mathematic, it is the base for many fields like Algebra, Topology, and Probability it gives also an essential foundation to the development of many concepts like infinity and many other sophisticated mathematical concepts. set theory is used beyond that, it has numerous applications in other sciences like computer science and philosophy.

So, what is a set?! well, a set is a collection of things, objects, or items that we call the “members” or “elements” of the set., not that these objects may not be mathematical objects.

Usually, we note a set with a capital letter, a set may be defined by a formula or a rule for its elements or by listing its different elements between braces, here is an example:

Using a rule or formula: Let be the set of odd numbers between and .

Using the listing way: .

As we mentioned a set can contain non-mathematical objects like .

There are two types of sets: Finite and Infinite.

For a finite set, all the elements can be listed but in the case of an infinite set since as their name suggests they have an infinite number of elements, then we can’t write an explicit list of all elements, instead the set needs to be defined by a rule or a formula to specify its members.

The empty set, as its name suggests, is a set that has no elements, we note it by two braces with no elements inside or by the symbol.

**Examples of sets:**

- The natural numbers .
- The natural odd numbers .
- The natural even numbers .
- The prime number .
- The natural numbers between 147 and 506 .

For the mathematical writing we define a set by stating the rules or properties that the elements satisfy, we note:

or

and we read: is the set containing every element that satisfies the rule …

or is the set of the elements such as satisfies the property …

**Examples:**

- The natural numbers .
- The natural odd numbers .
- The natural even numbers .
- The natural numbers between 147 and 506 .
- .

Let’s suppose an object and two sets and .

If the object is an element or an item of the set (or in other terms: if contains ) then we note .

If every element of is contained in (or in other terms if any element of is an element of ), then we say that is a subset of or we can say is included in and we note .

**Examples:**

1- and are two set defined as follow:

& , we have all the elements of are also elements of be and therefore is a subset of : ;

2- ;

;

Note that this relation of inclusion is transitive, meaning that if is a subset of and if is a subset of , then is a subset of .

**Example:** we have and we have and we have , therefore, i.e., the set of natural numbers is a subset of the set of real numbers.

In Set theory, it is possible to do operations on sets, there are many possible operations, here is a list of the main operations on sets:

**The union of sets:**

The union of and is the set containing both elements from and (including the elements that belong to both and ), we note and we read union .

Mathematically we write: .

**Example:** let and be defined as follow: ; .

The union of and is the set .

**The intersection of sets:**

The intersection of and is the set containing only the elements that belong to both sets and . We note and we read The intersection of and or the intersection of with .

Mathematically we write: .

**Example:** let and be defined as follow: ; .

The intersection of and is the set .

**The difference of sets:**

The difference of and is the set that contains only the elements of that are not members of . We note (and sometimes denoted ) and we read The difference between and (or minus ).

Mathematically we write: .

Note that is equal to the union of and .

**Example:** let and be defined as follow: ; .

The difference between and is .

**The symmetric difference of sets:**

The symmetric difference of and is the set that contains the elements that are from and don’t belong to and the elements from that don’t belong to , in other terms, it contains the elements that are members of only or . We note or and we read The symmetric difference of and .

Mathematically we write: .

Note that: The symmetric difference of and is the union of and .

i.e., .

It is also the difference of and .

i.e., .

**Example:** let and be defined as follow: ; .

The symmetric difference of and is .

**The Cartesian product of sets:**

The Cartesian product of and is the set that contains all the possible orders pairs such as is an element of and is an element of . We note and we read: The cartesian product of and ( times ).

Mathematically we write: .

**Example:** let and be defined as follow: and .

The Cartesian product of and is

.

**The power set of a set:**

The power set of is the set that contains all the possible subsets of . We note and we read The power set of .

**Example:** let be defined as follow: ; The power set of is (note that is a subset of ).

The different operations on sets have their properties that are useful for easier handling and manipulation, here are the main properties of the operations on sets:

**Associativity: **

We have the associativity properties for both the union and the intersection of sets, it goes as follow:

- The union of the with the set union of and is equal to the union of the set union of and with the set . It may be written mathematically as follow:

- The intersection of the with the set intersection of and is equal to the intersection of the set intersection of and with the set . It may be written mathematically as follow:

**Commutativity:**

We have the commutativity properties for both the union and the intersection of sets, it goes as follow:

- The union of and is equal to the union of and . It may be written mathematically as follow:

- The intersection of and is equal to the intersection of and . It may be written mathematically as follow:

**Distributivity: **

We have the associativity properties for the union with the intersection of sets, it goes as follow:

- The union of with the intersection of and is equal to the intersection of the union of and with the union of and . It may be written mathematically as follow:

- The intersection of and the union of and is equal to the union of the intersection of and and the intersection of and . It may be written mathematically as follow:

**Idempotency:**

In mathematics idempotency is a property of certain operations, it means that if we apply the same operation many times there will be no changing of the result. We have the idempotency property for the union and the intersection, it goes as follow:

- The union of with (with itself) equals :

- The intersection of with (with itself) equals :

**Other properties:**

If is a subset of (), then we have:

In this article we learned a little bit about sets, their definition, and the different operations and their properties, it may be considered as an introduction to set theory, and since set theory is fundamental for many other branches like Algebra and probability, this introduction can help us understand and study other subjects with ease, but keep in mind this is just an introduction and there is a lot more to learn!

In the meantime, you can take a look at some articles about limits and infinity like Limit of a function: Introduction, Limits of a function: Operations and Properties, Limits of a function: Indeterminate Forms, or the ones about Infinity: Facts, mysteries, paradoxes, and beyond or check the one about Probability: Introduction to Probability Theory!!!!!

And don’t forget to join us on our Facebook page for any new articles and a lot more!!!!!

The post Set Theory: Introduction appeared first on Math Academy.

]]>