Basic Set Theory

Set Theory

Apart from numbers, let us work with the objects. We generally come across different types of collection or groups such as a group of colors, a group of food items, a collection of garments, a collection of books, etc. 

This collection or group of objects when mathematically represented is called a set. 

Sets are often represented in curly brackets {} and items of the set are called elements or entities.

The German mathematician and logician Georg Cantor created a ‘Theory of sets’ or ‘Set Theory’.

In this article let us study set theory definition, set theory basics, types of sets, set theory symbols, and set theory examples.

Set Theory Definition

Set theory is a branch of mathematics that deals with the collection of objects termed as sets.

To understand set theory let us consider an example: we will make a list of the color of rainbow.

Violet, indigo, blue, green, yellow, orange, red.

The above-mentioned list is a well-defined list of colors. Any name of fruits or flowers cannot be mentioned in this list. Such a well-defined list of any element is called the set.

If the same list is written in reverse as

Red, orange, yellow, green, blue indigo, violet.

The set does not alter, it remains the same, as the order of the elements does not matter in sets.

We write sets using curly braces {} and denote them with capital letters.

For Example,

A = {1,2,3,…,10} is the set of the first 10 counting numbers, or naturals, B = {Red, Blue, Green, Yellow} is the set of colors, N = {1,2,3, 4, 5…} is the set of all naturals, and Z = {...,−3,−2,−1,0,1,2,3,…} is the set of all integers.

Set Theory Rules

Sets should be represented by some rules they are:

  • Sets should be symbolized by capital alphabets like A, B, C, D, ……

  • Members of the set should be denoted by lower case alphabets like a,b,c,d, …..

  • The members belonging to the set is represented by the symbol ‘ε’ called as epsilon and read as “belongs to” for example if a is member of set B it is represented as a ε B and not belongs to is represented as

Set Representation

Sets can be represented in two ways

  • Roster Form ( Tabular form) and 

  • Set builder form

  • Roster Form

In roster form, the elements of the set are listed in curly braces and are separated by commas. This form is also called the list notation.

For example If A is a set containing the list of vowels. It will be represented as 

A = { a, e, i, o, u}. The order of the elements does not matter. It can also be represented as 

A = { e, i, a, 0 , u}.

  • Set Builder Form

In Set builder form all the elements have a common property. This property is not applicable to the objects that do not belong to the set. E.g. If set S has all the elements which are natural numbers less than 8, it is represented as:

S = {x|x is a natural number and x < 10}  

And it is read as  “the set of all x such that x is a natural number and is less than 10”  in place of x any alphabet can be used.

The roster form of this set would be

S = { 1, 2, 3, 4, 5, 6, 7, 8, 9}

Set Theory Symbols

Some of the basic set theory symbols are given below:

Set Theory Symbols chart

Symbol

Meaning

{}

Represents the collection of elements of the set

Example { a, b, c, d,....}

ɸ

Empty set. It contains no element.

ɸ={}

ε

Member of the set belongs to the set

Example: a ε A

N

Represents the set of all Natural numbers i.e. all the positive integers.

This can also be represented by 

Examples: N = ( 1, 2, 3, 4, 5, 6, 7, 8, .. ….}

Z

Represents the set of all integers

Positive and negative integers are denoted by 

Z+ and Z-  respectively.

Examples: Z = { …….-3, -2, -1, 0, 1, 2, 3, ……..}

Q

Represents the set of Rational numbers

Positive and negative rational numbers are denoted by 

Q+  and Q-  respectively.

 Example : Q= {x | x=a/b, a,b∈Z}

Q = { ½’ 2/4, 6/5…..}

R

Represents the Real numbers i.e. all the numbers located on the number line.

Positive and negative real numbers are denoted by 

R+ and R- respectively.

Examples: 1.2, 4.5, 7.8777

C

Represents the set of Complex numbers.

Examples: 6+ 2i, i, etc.


Types of Sets

Set theory is classified into different types of sets they are:

  • Finite set

  • Infinite set

  • Empty set

  • Singleton set

  • Equal set

  • Equivalent set

  • Power set

  • Universal set

  • Subset

Solved Examples

Set Theory Examples

  1. P is the set which contains all odd numbers less than 10

Solution: P= { 1, 3, 5, 7, 9}

P = {x | x is the odd numbers less than 10}


  1. D is the set of all even prime numbers.

Solution: D = { 2 }

Quiz Time

Set Theory Examples to Practice

  1. X is the set of multiples of 3 

  2. Y is the set of multiples of 6

  3. Z is the set of multiples of 9

FAQ (Frequently Asked Questions)

1. What is a Subset in Maths?

Consider a set of 5 natural numbers.

set {1, 2, 3, 4, 5}

A subset of this is {1, 2, 3}.

From this example, we can define a subset as a set of members or elements which is present in another set or a parent set.

Example: If Set A has {1,2,3} and Set B has {1, 2, 3, 4, 5}, then A is the subset of B because every element of set A is also present in set B.

So, we can say that  Set A is a subset of Set B if and only if every element of A is in B.

But if Set A has { 1,2,6 } then Set A is not a subset of B because the element 6 is not present in the parent Set B.

A subset is represented by the symbol ⊆and is read as ‘is a subset of’. In the above and example subsets are represented as A ⊆B, and read as Set A is a subset of Set B.

And ‘is not a subset of’ is represented by the symbol ⊈ .

2. What are the Proper Subsets?

Answer : A proper subset is one that contains few elements but not every element of the parent set whereas an improper subset contains every element of the parent set along with the null set. 

The null set or empty set is denoted by Φ or {}

For example, if Set A = {1, 2, 3}, then,

Number of subsets: {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} and Φ or {}.

Proper Subset: {1}, {2}, {3}, {1,2}, {2,3}, {1,3}

Improper Subset: {1,2,3} and Φ

Example : Set A = { 1,2,3,4,5 } and Set B = { 1, 2, 3, 4,5,6,7,8}

Set A is a proper subset of Set B as some of the elements is not present in Set A

Set A is considered to be a proper subset of Set B, if Set B contains at least one element that is not present in Set A.

A proper subset is denoted by ⊂ and is read as ‘is a proper subset of’ A ⊂ B.