# Intersection of Sets

## Definition of Intersection of sets

The Intersection of two Sets $A$ and $B$ may be defined as the collection of all the elements which are common in both the sets $A$ and $B$.

## Symbol of Intersection of sets

Intersection of two Sets $A$ and $B$ can be denoted by $A ∩ B$.

## Example of Intersection of Sets

To understand the concept of Intersection of two Sets, let us consider two Sets $A$ and $B$ such that,

$A = \{1, 2, 3, 4\}$

$B = \{3, 4, 5, 6\}$

The Intersection of the above two sets $A$ and $B$ will contain all the elements which are common in both the sets $A$ and $B$. Here, the elements, $3$ and $4$ are common in both the sets $A$ and $B$. Therefore,

$A ∩ B = \{3, 4\}$

The Venn Diagram for Intersection of the above two sets is given by,

## Basic Properties of Intersection

### Cumulative Property of Intersection

The Cumulative Property of Intersection of two sets $A$ and $B$ is defined as,

$A ∩ B = B ∩ A$

#### Explanation of Cumulative Property

To understand the Cumulative Property of Intersection of Sets, let us consider two sets $A$ and $B$ such that,

$A = \{1, 2, 3, 4\}$

$B = \{3, 4, 5, 6\}$

The Intersection of $A$ and $B$ will be,

$A ∩ B = \{3, 4\}$

The Intersection of $B$ and $A$ will be,

$B ∩ A = \{3, 4\}$

Therefore,

$A ∩ B = B ∩ A$

This property of Intersection is called the The Cumulative Property of Intersection.

### Associative Property of Intersection

The Associative Property of Intersection for three sets $A,\, B$ and $C$ may be defined as,

$A ∩ (B ∩ C) = (A ∩ B) ∩ C\,\,\,..............(i)$

#### Explanation of Associative Property

To understand the Associative Property of Intersection, let us consider three sets $A$, $B$ and $C$ such that,

$A = \{1, 2, 3\}$

$B = \{2, 3, 4\}$

$C = \{3, 4, 5\}$

The Left-Hand Side of the Equation $(i)$ will be,

$(B ∩ C)=\{3, 4\}$

$A ∩ (B ∩ C)=\{1, 2, 3\} ∩ \{3, 4\}$

$A ∩ (B ∩ C)=\{3\}$

Similarly, the Right-Hand Side of the Equation $(i)$ will be,

$(A ∩ B)=\{2, 3\}$

$(A ∩ B) ∩ C = \{2, 3\} ∩ \{3, 4, 5\}$

$(A ∩ B) ∩ C= \{3\}$

Therefore,

$A ∩ (B ∩ C) = (A ∩ B) ∩ C$

This Property of Intersection is called the Associative Property of Intersection.

### Idempotent property of Intersection

The Idempotent Property of Intersection states that the Intersection of a set $A$, with itself, is set $A$ only. Mathematically,

$A ∩ A = A$

#### Explanation of Idempotent Property

To understand the Idempotent Property of Intersection, let us consider a set $A$ such that,

$A = \{1, 2, 3, 4, 5\}$

Now, the Intersection of Set $A$ with itself will be,

$A ∩ A = \{1, 2, 3, 4, 5\} ∩ \{1, 2, 3, 4, 5\}$

$A ∩ A= \{1, 2, 3, 4, 5\}$

Therefore,

$A ∩ A= A$

This Property of Intersection is called the Idempotent Property of Intersection.

### Identity Property of Intersection

The Identity Property of Intersection states that the Intersection of any set $A$ with an empty set $∅$ is the empty set $∅$. Mathematically,

$A ∩ ∅ = ∅$

#### Explanation of Identity Property

To understand the Identity Property of Intersection, let us consider a set $A$ such that,

$A = \{1, 2, 3, 4, 5\}$

And the Empty set is given by,

$∅ = \{\}$

The Intersection of $A$ with the Empty Set will be,

$A ∩ ∅ = \{1, 2, 3, 4, 5\} ∩ \{\}$

$A ∩ ∅= \{\}$

Therefore,

$A ∩ ∅= ∅$

This Property of Intersection is called the Identity Property of Intersection.

## Solved Examples on Intersection

1. If $A=\{a, b, c, d\}$ and $B=\{c, d, e, f\}$ then, find $A ∩ B$?

Solution: Given, $A=\{a, b, c, d\}$ and $B=\{c, d, e, f\}$
$A ∩ B$ will have the elements that are common in both sets $A$ and $B$.
Therefore,
$A ∩ B=\{c, d\}$

2. If $A = \{ x|x \in N\}$ and $B = \{ x|x \in W\}$ then, find $A ∩ B$?

Solution: Given,
$A = \{ x|x \in N\}$ is the set of Natural Numbers.
$B = \{ x|x \in W\}$ is the set of Whole Numbers.

The Natural Numbers starts from $1$ and ends in infinity and the Whole Numbers starts from $0$ and ends in infinity. Therefore, $A ∩ B$ will contain the elements present in the set of Natural Numbers as they are common to both sets.
$A ∩ B=\{ x|x \in N\}$

3. If $A=\{x, y, z\}$, $B=\{y, z\}$ and $C=\{z, a, b\}$ then, find $A ∩ B∩ C$?

Solution: Given, $A=\{x, y, z\}$, $B=\{y, z\}$ and $C=\{z, a, b\}$.
Only the element $z$ is common in all the three sets. Therefore,
$A ∩ B∩ C=\{z\}$

4. If $A=\{a, b, c\}$, $B=\{d, e\}$ and $C=\{f, g\}$ then, find $A ∩ B∩ C$?

Solution: Given, $A=\{a, b, c\}$, $B=\{d, e\}$ and $C=\{f, g\}$
In the above three sets, there is no common element present. Therefore, the Intersection of the three sets will be the empty set $∅$.
$A ∩ B∩ C=\{∅\}$