Intersection of sets – Venn diagram

In this chapter we will learn to represent intersection of sets of using Venn diagram with example.

Let’s first review the intersection operation in set theory.

What is Intersection in set theory ?

If A & B are two sets, then intersection operation will result in elements common in set A & B.

For Example;
If A = { 2, 4, 6, 8, 10 }
B = { 1, 3, 6, 7, 10 }

Note that in above two sets, element 6 & 10 are common.

So, A intersection B = { 6, 10 }

Symbol of Intersection of Set

The intersection operation is represented by symbol ” ∩ “

So the intersection of set A & B can be expressed as “ A ∩ B

Set Intersection – General expression

In set builder form, the intersection of set A & B can be expressed as follows;

A ∩ B = { x : x ∈ A & B )

Th expression says that A intersection B consists of entity x, where x belongs to both A & B.

Using Venn diagram in Intersection operation

If you want to understand the basics of Venn diagram in set theory, please click the link.

In this topic we will graphically represent set intersection using Venn diagram.

Suppose A & B are two sets, the Venn diagram of intersection of set A & B is given below (i.e. A ∩ B )

Set intersection Venn diagram

In the above image;

Rectangular box represents universal set U

Two circles A & B represent set A & B.

⟹ The green area shows the common element between set A & B which represents A ∩ B

Hence the common green area between set A & B represents intersection of set A & B

Representing intersection operations with Venn diagram

(01) If A ⊆ B, then A ∩ B = A

It says that if A is subset of B, then intersection of set A & B will result in set A

Representing intersection of set graphically

In the above image;

⟹ Rectangle represents Universal set

⟹ Since set A is subset of B, the circle A lies inside circle B.

⟹ the green area represents A ∩ B

From the image you can observe that if A is subset of B, the intersection of set A & B will result in set A.
i.e. A ∩ B = A

(02) Commutative Property of intersection
A ∩ B = B ∩ A

The property says that in intersection operation, if we change the order of sets, the final result will still be the same.

Representing property in Venn diagram.

Intersection of two sets Venn diagram

In both the above image, the green area represent the intersection of set A & B.

Note that even after interchanging the position of set A & B, the intersection area (green area) remains the same.

(03) Given below are set A & B. Find the intersection of both sets and represent in Venn diagram.
A = { 5, 8, 10, 12, 15, 16, 19 }
B = { 6, 9, 10, 15, 17, 19 }

For intersection of set A & B, find the common elements between them.

A ∩ B = {10, 15, 19}

Given below is the representation in Venn diagram.

Venn diagram of A intersection B intersection C

In the above figure, the yellow area represent the intersection of set A & B.

(04) Given below are three sets A, B & C. Find the intersection of all the sets A ∩ B ∩ C.

A = { 2, 4, 6, 8 }
B = { 6, 8, 10, 12}
C = { 2, 3, 6, 9 }

The intersection of set A, B, C will result in element which is common in all of the three sets.

You can note that number 3 is present in all the sets A, B & C.

Hence, A ∩ B ∩ C = { 3 }

Representing the above intersection in Venn diagram.

Set Intersection Venn diagram

In the above image;

⟹ set A, B & C are represented by circles.

⟹ the green area is the common area of all the sets and therefore represents the intersection of set A, B & C.

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