Set Operations are performed on two or more sets to obtain the combination of elements based on the Operation. There are three major types of Operations performed on Sets like Intersection, Union, Difference in **Set Theory**. Check out Representation of Intersection of Sets using Venn Diagram, Properties of Intersection of Sets, Solved Examples in the later modules.

## Intersection of Sets Definition

Intersection of Sets A and B is the Set that includes all the Elements that are Common to Sets A and B. Intersection is represented using the symbol ‘∩’. All the elements that belong to both A and B denote the Intersection of A and B.

A ∩ B = {x : x ∈ A and x ∈ B}

If you have n sets i.e. A_{1}, A_{2}, A_{3}…..A_{n} all are Subsets of Universal Set U the intersection is the set of elements that are in common to n sets.

### Intersection of Sets Venn Diagram

Consider Two Sets A and B and their Intersection is depicted pictorially using the following Venn Diagram. A, B are subsets in the Universal Set. Intersection of Sets is all those elements that belong to both the Sets A and B. Shaded Portion denotes the Intersection of Sets A and B. Intersection of Sets A and B is represented as A ∩ B and is read as A Intersection B or Intersection of A and B.

A ∩ B = {x : x ∈ A and x ∈ B}.

Clearly, x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

Thus, from the Definition of Intersection, we can conclude that A ∩ B ⊆ A, A ∩ B ⊆ B.

### Properties of Intersection of Sets

(i) A ∩ A = A (Idempotent theorem)

(ii) A ∩ B = B ∩ A (Commutative theorem)

(iii) A ∩ U = A (Theorem of Union)

(iv) A ∩ ϕ = ϕ (Theorem of ϕ)

(v) A ∩ A’ = ϕ (Theorem of ϕ)

(vi) If A ⊆ B, then A ∩ B = A.

### Solved Examples on Intersection of Sets using Venn Diagram

1. If A = {a, b, d, e, g, h} B = {b, c, e, f, h, i, j}. Find A ∩ B using the Venn Diagram?

Solution:

Given Sets are A = {a, b, d, e, g, h} B = {b, c, e, f, h, i, j}

Draw the Venn Diagram for the given Sets and then find the Intersection of Sets.

Intersection is nothing but the elements that are common in both Sets A and B.

A ∩ B = { b, e, h}

2. If C = { 3, 5, 7} D = { 7, 9, 11}. Find C ∩ D using Venn Diagram?

Solution:

Given Sets are C = { 3, 5, 7} D = { 7, 9, 11}

Let us represent the given sets in the diagrammatic representation of sets

Intersection is nothing but the elements that are common in both Sets C and D.

C ∩ D = {7}