Set Theory: Introduction

by | Oct 11, 2021 | Math Learning

Set theory is one branch of mathematics that is concerned with the study of sets and their properties, size, and their inter-relations (like intersections and unions), in this article we will learn about set theory, the different operations on sets and cardinality, with some illustrating examples.

Introduction:

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 A be the set of odd numbers between 12 and 26.

Using the listing way: A=\lbrace 13,15,17,19,21,23,25\rbrace.

As we mentioned a set can contain non-mathematical objects like B=\lbrace Pen, Tree, Water, toy, Car\rbrace.

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 \mathbb{N}=\lbrace 0,1,2,3,4,... \rbrace.
  • The natural odd numbers O=\lbrace 1,3,5,7,9,... \rbrace.
  • The natural even numbers E=\lbrace 0,2,4,8,10,... \rbrace.
  • The prime number P=\lbrace 2,3,5,7,11,... \rbrace.
  • The natural numbers between 147 and 506 A=\lbrace 148,149,150,151,152,... \rbrace.

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

A=\lbrace x\;  : \;  x \text{ satisfies the rule ... }\rbrace or B=\lbrace x \; | \; x \text{ satisfies the property ...}\rbrace

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

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

Examples:

  • The natural numbers N=\lbrace x : x\in\mathbb{N}\rbrace.
  • The natural odd numbers O=\lbrace x \; | \; x=2k+1, k\in\mathbb{N}\rbrace.
  • The natural even numbers E=\lbrace x \; : \; x=2k, k\in\mathbb{N}\rbrace.
  • The natural numbers between 147 and 506 A=\lbrace x \; | \; 147<x<506, \; x\in\mathbb{N}\rbrace.
  • B=\lbrace x \;: \; x=x^{2}, \; x\in\mathbb{N}\rbrace.

 

Relations between the sets:

Let’s suppose an object o and two sets A and B.

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

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

Examples:

1- A and B are two set defined as follow:

A=\lbrace 14,5,3,100,2\rbrace & B=\lbrace 25,14,38,17,5,3,0,158,128,66,100,31,38,2\rbrace, we have all the elements of A are also elements of be and therefore A is a subset of B: A\subseteq B;

2- \mathbb{N}\subseteq \mathbb{Z};

\mathbb{Z}\subseteq \mathbb{Q};

\mathbb{Q}\subseteq \mathbb{R}

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

Example: we have \mathbb{N}\subseteq \mathbb{Z} and we have \mathbb{Z}\subseteq \mathbb{Q} and we have \mathbb{Q}\subseteq \mathbb{R}, therefore, \mathbb{N}\subseteq \mathbb{R} i.e., the set of natural numbers is a subset of the set of real numbers.

 

Operations on sets

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 A and B is the set containing both elements from A and B (including the elements that belong to both A and B), we note A\cup B and we read A union B.

Mathematically we write: A\cup B=\lbrace x \; : \; x \in A, \text{ or } x \in B\rbrace.

Example: let A and B be defined as follow: A=\lbrace 1,2,3,5,7,10\rbrace; B=\lbrace 0,2,4,6,8,10\rbrace.

The union of A and B is the set A\cup B=\lbrace 0,1,2,3,4,5,6,7,8,10\rbrace.

 

The intersection of sets:

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

Mathematically we write: A\cap B=\lbrace x \; : \; x \in A, \text{ and } x \in B\rbrace.

Example: let A and B be defined as follow: A=\lbrace 1,2,3,5,7,10\rbrace; B=\lbrace 0,2,4,6,8,10\rbrace.

The intersection of A and B is the set A\cap B=\lbrace 2,10\rbrace.

 

The difference of sets:

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

Mathematically we write: A\setminus B=\lbrace x \; : \; x \in A, \test{ and } x \not\in B\rbrace.

Note that A is equal to the union of A\cap B and A\setminus B.

Example: let A and B be defined as follow: A=\lbrace 1,2,3,5,7,10\rbrace; B=\lbrace 0,2,4,6,8,10\rbrace.

The difference between A and B is A\setminus B=\lbrace 1,3,5,7\rbrace.

 

The symmetric difference of sets:

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

Mathematically we write: A\bigtriangleup B=\lbrace x \; : \; (x \in A) \; \oplus\; (x \in B)\rbrace=\lbrace x \; : \; [x \in A, \text{ and } x \not\in B] \text{ or } [x \not\in A, \text{ and } x \in B]\rbrace.

Note that: The symmetric difference of A and B is the union of A\setminus B and B\setminus A.

i.e., A\bigtriangleup B=A\setminus B \cup B\setminus A.

It is also the difference of A\cup B and A\cap B.

i.e., A\ominus B= (A\cup B)\(A\cap B).

Example: let A and B be defined as follow: A=\lbrace 1,2,3,5,7,10\rbrace; B=\lbrace 0,2,4,6,8,10\rbrace.

The symmetric difference of A and B is A\bigtriangleup B=\lbrace 1,3,5,7,0,4,6,8\rbrace.

 

The Cartesian product of sets:

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

Mathematically we write: A\times B=\lbrace (a,b) \; : \; a \in A, \text{ and } b \in B\rbrace.

Example: let A and B be defined as follow: A=\lbrace 3,7\rbrace and B=\lbrace Algebra, Calculus, Probability\rbrace.

The Cartesian product of A and B is

    \[A\times B=\lbrace (3,Algebra),(3,Calculus),(3,Probability),(7,Algebra),(7,Calculus),(7,Probability) \rbrace\]

.

 

The power set of a set:

The power set of A is the set that contains all the possible subsets of A. We note \mathcal{P}(A) and we read The power set of A.

Example: let A be defined as follow: A=\lbrace 3,7\rbrace; The power set of A is \mathcal{P}(A)= \lbrace \lbrace \rbrace, \lbrace 3\rbrace, \lbrace 7\rbrace, \lbrace 3,7\rbrace \rbrace (note that A is a subset of A).

 

The properties of the operations on sets

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 A with the set union of B and C is equal to the union of the set union of A and B with the set C. It may be written mathematically as follow:

    \[\(A \cup (B\cup C)=(A\cup B)\cup C\)\]

  • The intersection of the A with the set intersection of B and C is equal to the intersection of the set intersection of A and B with the set C. It may be written mathematically as follow:

    \[\(A \cap (B\cap C)=(A\cap B)\cap C\)\]

 

Commutativity:

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

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

    \[\(A \cup B=B\cup A\)\]

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

    \[\(A \cap B=B\cap A\)\]

 

Distributivity:

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

  • The union of A with the intersection of B and C is equal to the intersection of the union of A and B with the union of A and C. It may be written mathematically as follow:

    \[\(A \cup (B\cap C)=(A\cup B)\cap (A\cup C)\)\]

  • The intersection of A and the union of B and C is equal to the union of the intersection of A and B and the intersection of A and C. It may be written mathematically as follow:

    \[\(A \cap (B\cup C)=(A\cap B)\cup (A\cap C)\)\]

 

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 A with A (with itself) equals A:

    \[A \cup A=A\]

  • The intersection of A with A (with itself) equals A:

    \[A \cap A=A\]

 

Other properties:

    \[A \cup \emptyset = A\]

    \[A \cap \emptyset = \emptyset\]

    \[A-A=\emptyset\]

If A is a subset of B (A\subseteq B), then we have:

    \[A\cup B= A\cup (B-A)=B\]

    \[A\cap B=A\]

 

Conclusion:

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!!!!!

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!!!!!

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