Set Theory | Concept of Subset, Union and Intersection PDF

Set Theory: Are you getting confused in the set theory concept? Get 100% clarification of set theory, Subset, Union, Intersection and Venn Diagram with PDF.

The concept of set Theory

Set Theory

Introduction to Set Theory

The word ‘set‘ and other similar words are frequently used in our daily life’s vocabulary, such as a set of cards, a bunch of keys, a pack of cigarette and so on. Such phrases express the common notation of ‘set’.

Set is the definite collection of well-defined objects is called set. These well-defined objects are called Elements or members of a set.

Every object of a set is known as an element of that set.

Example: A={ b, h, a, r, t, i}

Where b, h, a, r, t, i is called elements of that set

Important Notations in set Theory

  • { a,b}: Used at the starting and ending of a set.
  • ∈: Belongs to
  • ∉: Not Belongs to
  • ∅: Fi (Empty set/ Nothing/ Null set/ { })
  • : -such that
  • ⊂: Subset
  • ⊃: Super Set
  • ⊆: Subset Equals to
  • ⊇: Superset Equals to
  • ∪: Union
  • ∩: Intersection
  • Δ: Symmetric Difference
  • ‘ : Complement
  • ⇔: Equivalent to

Representation of sets

Tabular or Roaster Method

In this method, we put all the elements within curly braces { } by making a list.


  1. B= { mango, banana, apple}
  2. A= {1, 2, 3, 4, 5, 6, 7}

Set Builder Method

In this method, we list the property or properties, satisfied by the elements of sets.


  1. B= {x: x ∈ set of Fruits}
  2. A= {x: x∈ N and x ≤ 6}= {x: x is a natural number less than or equals to six}

Related Post: Trigonometric Formulas

Types of set Theory

In the concept of set theory, we have 12 types of Set. They are as follows:

  1. Empty Set
  2. Singleton Set
  3. Finite Set
  4. Cardinal number of finite Set
  5. Infinite Set
  6. Equivalent Set
  7. Equal Set
  8. Subset
  9. Proper Subset
  10. Universal Set
  11. Power Set
  12. Complement Set

Let us discuss one by one

1) Empty Set/ Null Set/ Fi Set

A set is said to be null set or empty set or fi set, it its has no element. It is denoted by Ø or { }

Example: A= {x: x ∈ N and f(2x²+1)=0}= Ø

2) Singleton Set

A set is said to be Singleton set if it has only one element inside it.


  1. A= {x: x ∈ N and 3x – 9 = 0}
  2. B= { mango}

3) Finite Set

A set is said to be finite if it has a finite number of elements.


  1. A= {x: x∈ N and x< 103}
  2. B= { a, b. c, d, e,f, ….p, q,r}

4) Cardinal number of finite set

Cardinal Number of a finite set can be calculated by counting the number of  elements inside the set.

Example: A= { 20, 30, 40, 50, 60}, then n(A)= 5

5) Infinite Set

A set is said to be infinite set if the number of elements are infinite or un-countable.

Example: A= {1, 2, 3, 4, 5, 6, 7, ……}

6) Equivalent Sets

Two sets are said to be equivalent if the cardinal number of two sets are same.

Example: Let A= {2, 6, 5, 9} and B= {p, a, u, j}

n(A)= 4 and n(B)= 4 So, A and B are equivalent sets

7) Equal Set

If two sets have the same elements, then the two sets will be equal sets. Here mandatory is not that they are in the same order.

Example: A= {r, a, j, p} and B= {p, a, j, r}

Here A and B have same elements r, a, j, p but they are in different order.

Still A=B

8) Subset

Let A, B be sets. We say that A is a subset of B, or B is super subset of A if every element of A is an element of B and can be written as A⊂B as subset and B ⊃A super subset.

i.e x∈ A ⇒x∈ B

Example: A= {1, 5, 7}, B= {1, 3, 5, 7, 9}

Here the elements of  A i.e, 1, 5, 7 are also an element of set B. Then A ⊂B or B ⊃ A

If the element of A is not a subset of B then we can say that A ⊄B.


  • Every set is a subset of itself.
  • Empty set Ø is a subset of every set.

9) Proper Subset

If every element of A is an element of B  but there is at least one element in B which is not in A, then A is a proper subset of B.

Example: A= {a, b} and B= { b, c, a, d}


10) Universal Set

A super set of each set is known as universal set. It is denoted by U.

11) Power Set

Let A be a set, then the collection of all subsets of A is known as Power set of A. It is denoted by P(A).

Example:  A= { a, b, c, d}, then 

P(A)= { Ø, {a}, {b}, {c}, {d}, {a, b}, {b, c}, {c, d}, {d, a}, {a, b, c, d}}

12) Complement Set

The complement of a set A is the set of elements which belong to the universal set and do not belong to set A. It is denoted by A’.

Example: A= {p, q, r, s, t, u, v} and B= {p, q, r, s, t}

So A’= { u, v}

Operations on Set Theory

Union of Sets

The Union of two sets A and B is the set of elements which belong to A or B or to both A and B. It is denoted by A∪B.

Example: A= {1, 5, 6} and B= {2, 5, 7. 8}

So, A ∪ B= {1, 2, 5, 7, 8}

Venn Diagram of Union

VennDIagram Union Set Theory

Intersection of Sets

Intersection of two sets A and B is the set of elements common to both A and B. It is denoted by A ∩ B.

Example: Let A= {m, n, o, p} and B= {p, q, r, s} then A ∩ B= {p}

Venn Diagram of Intersection

VennDIagram Set Theory intersection

Difference of Sets

Let A and B are two non-empty sets, then their difference is A-B or B-A.

A-B is a set of elements of A, which do not belong to B and B-A is a set of elements of B, which is not belonging to A.

Example: A= { 8, 9, 10, 11} and B= { 9, 11, 13, 12}

then A-B= { 8, 10,} and B-A= { 12, 13}

Venn Diagram for Difference of Set

A-B SET THEORY Venn diagram

Symmetric Difference of sets

Let A and B are two sets, then the union of A-B and B-A is known as symmetric difference of sets A and B. It is denoted  by A Δ B= (A-B) ∪ (B-A)

Set Theory PDF Download

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