# Intersection of Sets

## What is Intersection of sets?

If A & B are two given sets, then the intersection of the given sets will contain elements which are common in set A & B.

For Example;
Given below are sets A & B;

Here the intersection of set A & B will contain common elements between A & B.

Note that elements 10 & 15 are the common elements present in both set A & B.

Hence for A intersection B, we get the following elements.

Conclusion
In set intersection operation, we look for elements common in the given sets.

## Representing Set Intersection

The intersection operation between two set is represented by symbol ” ∩ ”

Hence, the intersection of set A & B is expressed as follows;

Note that we can do intersection operation in three or more sets.

If A, B & C are the given sets, then intersection of the sets is expressed as:

### Representing set intersection through Venn Diagram

We know that Venn diagram is used for graphical representation of different sets.

In Venn diagram, the universal set is represented through rectangular box and sets are represented through circles.

Now consider two sets A & B with following elements;
A = { 5, 7, 9, 11, 13 }
B = { 4, 7, 10, 13, 15 }

Below is the Venn diagram for intersection of set A & B.

In the above Venn diagram, the grey area in the middle represents the intersection of sets A & B.

A ∩ B = { 7, 13 }

## Intersection of three sets

As i have already mentioned above, the intersection operation is not limited to two sets.

You can easily do the intersection of three or more sets using the above process.

Let there are three sets A, B & C.
The intersection of A, B & C will contain elements which are common in the set A, B & C.

For Example
Given below are three sets A, B & C
A = { 5, 7, 9, 13}
B = { 7, 15, 19, 22 }
C = { 4, 7, 16, 21 }

Find A ∩ B ∩ C.

Solution
Note that in all the given sets A, B & C, number 7 is the only common element.

Hence, A ∩ B ∩ C = 7

### Representing intersection of three sets in Venn Diagram

If three sets A, B & C are given, then Venn diagram is represented as;

Here the three circles represent the set A, B & C.

The blue area in the middle represents the intersection of three sets, A ∩ B ∩ C. It is the area which shows the common elements present in the given set.

## Solved Questions for Set Intersection

(01) Given below are three sets A, B & C.

Find:
(i) A ∩ B
(ii) B ∩ C
(iii) C ∩ A
(iv) A ∩ B ∩ C

Solution

(i) A ∩ B

To find intersection, note the common element between set A & B

A ∩ B = { 58, 59, 60 }

(ii) B ∩ C

Note the common element between B & C.

B ∩ C = { 59, 60, 61 }

(iii) C ∩ A

Write down the common elements between C & A.

C ∩ A = { 59, 60 }

(iv) A ∩ B ∩ C

Find the common elements between A, B & C.

A ∩ B ∩ C = { 59, 60 }

(02) Given below are two sets A & B.
A = { 22, 35, 41, 49, 53 }
B = { 𝜙 }

Find A intersection B ( A ∩ B)

Solution

Note that the set B is a null set.
It means that set B doesn’t contain any elements.

Observe that there is no common element between the given set A & B.

Hence, A ∩ B = { }

(03) Given below are sets A & B such that;

Find A ∩ B

Solution
Let’s convert both the sets into Roster form.

Given above is the set name A.
Set A contains element x such that 4 < x < 8.

Set A can be written as:
A = { 5, 6, 7 }

Similarly set B can be written as;
B = { 6, 7, 8, 9 }

For intersection operation, find the common element between set A & B.

A ∩ B = { 6, 7 }

(04) Given below are two sets A & B.

Find A ∩ B and also draw Venn diagram for above sets.

Solution
To find A ∩ B, locate the common elements between set A & B.

Note that elements Mango and Cherry are the common elements between the two.

Hence, A ∩ B = { Mango , Cherry }

Given below is the Venn diagram for sets A & B.

The colored portion between two circle represents A ∩ B

(05) Given below are two sets.

Find:
(i) A ∩ B
(ii) Venn Diagram.
(iii) Is set B subset of A?

Solution
(i) Between set A & B, elements 2, 4 & 6 are common.
Hence, A ∩ B = { 2, 4, 6 }

(ii) Given below is the Venn Diagram for set A & B.

In the above figure, the grey circle represents both set B and A ∩ B.

(iii) Set B is subset of A ?

Since all the elements of set B is present in A, set B is the subset of A