## 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=\{∅\}

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