What is Set Theory?

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

A set is defined as a collection or group of objects. Objects in a set are defined as elements. For example, a football team contains 11 players, and of those 11 players each player is defined as an element of the set representing the players in the team.

Sets and their Representations

Sets are usually represented in two ways:

1) Roster form

2) Set Builder Form

Roster Form

All the elements of a set in roster form are listed. The elements are separated by commas and are enclosed within braces { }. For example, the set of all odd positive integers less than 7 can be generated as:

The odd positive integers less than 7 are 1, 3, and 5.

According to the roster form, elements should be separated by commas so the required set can be represented as $A=\left\{1,3,5\right\}$.

1. According to the set theory, the order of elements is not important. The elements can be returned in any order.

For example, the set of all natural numbers which divide 24 is {2, 1, 3, 4, 12, 8, 6}.

2. While writing the set in roster form an element is not generally repeated, i.e., all the elements in the set are taken as distinct. For example, the set of letters forming the word ‘SEED’ is {S, E, D}.

Set Builder Form

All the elements of a set written in the set-builder form own a common property that is not owned by any element outside the set. This property is used to write the set.For example, in the set {a, e, i, o, u}, all the elements own a common property, namely, each of them is a vowel in the English alphabet, and no other letter owns this property. Denoting this set by V, we write

$V=\left\{x:xisavowelinEnglishalphabet\right\}$

Here x is the symbolic representation of elements in the set.

The symbol ‘:’ is read as such that.

The symbol ‘{}’ is read as the set of all.

The set $V=\left\{x:xisavowelinEnglishalphabet\right\}$ is read as “the set of all x such that x is a vowel in English Alphabet”.

Example:

$S=\left\{11,22,33,44,55,66,77,88,99\right\}$

This can be return set builder form as:

$S=\left\{p:pisasetoftwodigitmultiplesof11\right\}$

Types of Set

Empty set: A set containing no elements is called an empty set.

For example, $A=\left\{x:xisasetofevenprimelessthan2\right\}$. There is no even prime number less than 2, so set A ={} is an empty set or null set or void set.

Finite set: A set containing a finite number of elements is called a finite set.

Infinite set: A set containing an infinite number of elements is called an infinite set.

For example, the set of all natural numbers is an infinite set.

Equal set: Two sets are said to be equal if every element in the two sets are equal i.e., all the elements must be the same.

For example, $A=\left\{1,2,3,4,5\right\}B=\left\{2,3,4,5,1\right\}$.

A and B are equal sets because all elements of set A are present in set B and vice versa.

Equivalent set: Two sets are said to be equivalent whenever they have an equal number of elements.

For example, $A=\left\{2,4,6\right\}B=\left\{1,3,5\right\}$.

The number of elements in set A and set B is 3, so the sets are equivalent.

Subset: A set ‘A’ is a subset of set B if every element of set A is present in set B.

It is denoted as $A\subset B$.

The empty set is a subset of every set.

Power set: The set of every possible subset of a set is called its power set.

Universal Set: A set containing all the sets under any consideration is called a universal set.

Singleton set: A set containing only a single element is called a singleton set.

For example, A={1} is a singleton set.

Set Theory Symbols

• N : The set of all natural numbers (Ex:1 2 3 )
• Z : The set of all integers (ex:-2,-1,0,30)
• Q : The set of all rational numbers (-1/2,3/5)

• Z+ : The set of positive integers (1,2,3,30)
• Q+ : The set of positive rational numbers (1/2,3/5)
• R+:  The set of positive real numbers

Operation on Sets

Union of Sets: If A and B are two sets, then the union of A and B is the set that contains all the elements of sets A and B. The symbol ‘∪’ is used to denote the union.

Symbolically, it is represented as A $\cup$B and usually read as ‘A union B’.

For example, if A={1,2,3,4} and B={1,2,4,5,6,7}, then,

A $\cup$ B={1,2,3,4,5,6,7}If you observe A $\cup$

 B contains all elements of A and B but according to set rules each element appears only once.

Intersection of Sets: If A and B are two sets, then the intersection of A and B is the set that contains all the common elements of set A and set B.

The symbol ‘∩’ is used to denote the intersection.

For example, if A={1,2,3,4}, B={1,2,4,5,6,7}, then,

A $\cap$ B={1,2,4}

Difference of Sets: If A and B are two sets, then the difference of A and B is the set of elements that belong to set A and but not to set B.

Symbolically, we write A – B and it is read as “A minus B”.

For example, if A={1,2,3,4}, B={1,2,4,5,6,7}, then,

A $-$ B= {3}

Complement of a Set: If U is a universal set of set A, then the complement of set A is defined as the set of all the elements that are not present in set A.

A′ is the symbolic representation of compliment of set A.

For example, if U={1,2,3,4,,5,6,7,8}, A={1,2,3,4}, then,

A′={5,6,7,8}

Formulas

Consider that A, B and C are three sets, ‘φ’ represents null set  and U represents the universal set, then the following formulas hold:

• n(A U B) = n(A) + n(B), when A and B are disjoint sets.

• (A′)′ = A
• φ′ = U
• U′ = φ

Practice Problems

If U = {1,2,3,4,5,6,7,8,9,122,111}, A = {2, 3, 4, 5, 6} and B = {5, 6, 7,8, 9}, Then find (A – B)′.

(A – B)={2,3,4}

(A – B)′=U-( A – B)

(A – B)′={ 1,5,6,7,8,9,122,111}

If there are two sets X and Y such that their union has 60 elements, the set X has 34 elements and the set Y has 32 elements, how many elements does X ∩ Y have?

n(X $\cup$ Y) = n(X) + (n(Y) – n (X $\cap$ Y)

n(X)=34

n(Y)=32

n(X $\cup$ Y) =60

n(X $\cap$ Y)=34+32-60

n (X $\cap$ Y)=6

Context and Applications

This topic is significant in the professional an entrance exams for both undergraduate and graduate courses, especially for

• B.Sc. mathematics
• M.Sc. mathematics