# Intersection of Sets

The intersection of two sets *A* and *B *( denoted by *A∩B *) is the set of all elements that is common to both *A* and *B*. In mathematical form,

For two sets A and B, A∩B = { x: x∈A and x∈B }

Similarly for three sets A, B and C,

A∩B∩C = { x: x∈A and x∈B and x∈C }

## Intersection of Sets Examples

### Example #1: Intersection of Two Sets With Venn Diagram

**If A = {a, b, c, d, e} and B = {d, e, f, g}, find A∩B.**

Here, A = {a, b, c, d, e} B = {d, e, f, g} Now, A∩B = {a, b, c, d, e} ∩ {d, e, f, g} ∴ A∩B = {d, e}

### Example #2

**Suppose A = { x: x is an integer between 1 and 7} and B = { x: x is an integer between 4 and 9} then, find A∩****B.**

Here, A = {2, 3, 4, 5, 6} B = {5, 6, 7, 8} ∴ A∩B = {5, 6} = { x: x is an integer between 4 and 7}

### Example #3: Intersection of Disjoint Sets

**If A = {a, b, c} and B = {d, e, f, g}, find A∩****B.**

Here, A∪B = {a, b, c} ∩ {d, e, f, g} ∴ A∪B = {} = ϕ

### Example #4

**If A = { x: x is an integer} and B = { x: x is an even integer} then, find A∩B. **

Here, A = {..., -3, -2, -1, 0, 1, 2, ...} B = {..., -2, 0, 2, ...} ∴ A∩B = {..., -4, -2, 0, 2, ... } = {x: x is an even integer}

### Example #5: Intersection of Three Sets With Venn Diagram

**If A = {a, b, c, d, e}, B = {d, e, f, g} and C = {c, e, f, h, i}, find A∩B∩C. **

Here, A∩B∩C = {a, b, c, d, e} ∩ {d, e, f, g} ∩ {c, e, f, h, i} The common element among all three sets is 'e' ∴ A∩B = {e}

## Properties of Intersection of Sets

### 1. Commutative Property

If *A* and *B* are two sets then, **A∩B = B∩****A**

Suppose, A = {1, 2, 3, 4} B = {4, 3, 5} A∩B = {3, 4} B∩A = {4, 3} ∴ A∩B = B∩A

### 2. Associative Property

If *A*, *B* and *C* are three sets then*, ***A∩(B∩C)= (A∩B)∩C.**

Suppose, A = {1, 2, 3, 4} B = {3, 4, 5, 6} C = {3, 4, 1, 5, 8, 9} B∩C = {3, 4, 5} A∩B = {3, 4} A∩(B∩C) = {3, 4} (A∩B)∩C = {3, 4} ∴ A∩(B∩C) = (A∩B)∩C

### 3. Identity Property

The intersection of a set and the empty set is always the empty set, i.e, **A∩ϕ = ϕ**.

Suppose, A = {a, b, c} B = ϕ = { } A∩B = {1, 2, 3, } ∩ {} = {} = ϕ ∴ A∪ϕ = ϕ