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

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

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

Solution: Given, [latex]A = \{ x|x \in N\}[/latex] is the set of Natural Numbers. [latex]B = \{ x|x \in W\}[/latex] is the set of Whole Numbers. The Natural Numbers starts from [latex]1[/latex] and ends in infinity and the Whole Numbers starts from [latex]0[/latex] and ends in infinity. Therefore, [latex]A ∩ B[/latex] will contain the elements present in the set of Natural Numbers as they are common to both sets. [latex]A ∩ B=\{ x|x \in N\}[/latex]

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

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

Intersection of Sets | Definition Examples and Properties- Find Formula

Table of Contents

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 $A \cap B$ .

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,

Intersection of sets venn diagram — Intersection of Sets

Basic Properties of Intersection

Cumulative Property of Intersection

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

$A \cap B = B \cap A$

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,

$A \cap B = B \cap A$

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

Associative property of intersection venn diagram

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,

$A \cap (B \cap C) = (A \cap B) \cap C$

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,

$A \cap A = A$

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,

$A \cap A= A$

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 $\emptyset$ is the empty set $\emptyset$ . Mathematically,

$A \cap \emptyset = \emptyset$

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,

$A \cap \emptyset= \emptyset$

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 $A \cap B$ ?

Solution: Given, $A=\{a, b, c, d\}$ and $B=\{c, d, e, f\}$
$A \cap B$ 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 $A \cap B$ ?

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, $A \cap B$ 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 $A \cap B\cap C$ ?

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 $A \cap B\cap C$ ?

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 $\emptyset$ .
$A ∩ B∩ C=\{∅\}$

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Definition of Intersection of sets

Symbol of Intersection of sets

Example of Intersection of Sets

Basic Properties of Intersection

Cumulative Property of Intersection

Explanation of Cumulative Property

Associative Property of Intersection

Explanation of Associative Property

Idempotent property of Intersection

Explanation of Idempotent Property

Identity Property of Intersection

Explanation of Identity Property

Solved Examples on Intersection

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