In Set theory, there are various types of set operations like Subsets, Difference of sets, Union of sets, Complement of sets, and **Intersections of sets**. Students can easily understand and learn the set theory concepts by referring to our articles. So, today we have come up with the Intersection of Sets topic. Let’s go through this page and learn about the intersection of sets definition, symbol, properties, and examples.

## Intersection of Sets Definition

If the Intersection of two sets A and B, then it is denoted by A∩B, where it contains all the elements which are common to both A and B. The symbol to denote the intersection of sets is ‘∩’. All those elements that relate to both A and B denote the intersection of A and B. Hence, we can say that,

A ∩ B = {x : x ∈ A and x ∈ B}

For n sets A_{1}, A_{2}, A_{3},……A _{n }where all these sets are the subset of Universal set (set U). The definition of the intersection is the set of all elements which are common to all these n sets.

### What is the Intersection of Two Sets?

If there are two sets, Set A and Set B, then the intersection of sets is given by:

**A∩B = n(A) + n(B) – n(A∪B), **where

n(A) is the cardinal number of set A,

n(B) is the cardinal number of set B,

n(A∪B) is the cardinal number of a union of two sets ie., A and B.

**For Example:** The intersection of two sets ie., A = {1, 2, 3} and B = {2, 3, 4} then find A∩B?

We know,** A∩B = n(A) + n(B) – n(A∪B)**

After calculating the process of the intersection of two sets, the result is like this **A∩B = {2, 3}.**

### Properties of Intersection of a Set

i) **Commutative Law:** The union of two sets A and B follow the commutative law i.e., *A ∩ B = B ∩ A*

ii) **Associative Law:** The intersection operation follows the associative law i.e., If we have three sets A, B, and C then,* (A ∩ B) ∩ C = A ∩ (B ∩ C)*

iii) **Identity Law:** The intersection of an empty set with any set A gives the empty set itself i.e., *A ∩ ∅ = ∅*

iv)** Idempotent Law:** The intersection of any set A with itself gives the set A i.e., *A ∩ A = A*

v) **Law of U:** The intersection of a universal set U with its subset A gives the set A itself. *A ∩ U = A*

vi)** Distributive Law:** According to this law:

*A ∩ (B ∪ C) = ( A ∩ B ) ∪ (A ∩ C)*

### Solved Examples on Intersection of Sets

1. Let U be the universal set consisting of all the n – sided regular polygons where 5 ≤ n ≤ 9. If set A, B, and C are defined as:

A = {pentagon, hexagon, octagon}

B = {pentagon, hexagon, nonagon, heptagon}

C = {nonagon}

Find the intersection of the sets A and B

**Solution:**

Given Universal Set U = {pentagon , hexagon , heptagon , octagon , nonagon}, A = {pentagon, hexagon, octagon}, B = {pentagon, hexagon, nonagon, heptagon}, C = {nonagon}

Now, calculate the intersection of sets A and B

Here, the intersection of sets means the common elements consist in both A and B:

The elements of A = {pentagon, hexagon, octagon}, and B = {pentagon, hexagon, nonagon, heptagon}

Then, A ∩ B = {pentagon, hexagon}.

2. If A = {1,3,5,7,9} and B = {2,3,5,6,8}. Find intersection of two set A and B.

**Solution:**

Given Set A = {1,3,5,7,9} and Set B = {2,3,5,6,8}

Here 3, 5 are the common elements in both A and B sets.

So, the intersection of two sets A and B are,** A ∩ B = {3, 5}.**

3. If P = {x, y, z} and Q = {ф}. Find the intersection of two given sets P and Q.

**Solution:**

Given sets are P = {x, y, z} and Q = {ф}

The Intersection of given sets **P ∩ Q = { }.**