# Venn Diagrams in Set theory

In this chapter we will learn to use Venn diagram in set theory with solved examples.

## Using Venn Diagrams in set theory

Using Venn diagram we will graphically represent different sets in the form of pictures.

In Venn diagrams;
universal set U is represented by points in rectangular box

⟹ the subsets of U are represented by points inside the circle

Given below are important cases of sets in Venn diagram.

### Set A in Venn diagram

In the above figure;
⟹ Rectangle U represent universal set

⟹ Circle A represents set A which is subset of universal set.

Note;
We will consider every given set as subset of universal set unless it is said explicitly in the question.

### Set B is subset of A

In the above figure;

(a) Rectangle U represents universal set

(b) Circle A represent set A

(c) Circle B represent set B

(d) The presence of circle B inside circle A show that;

⟹ All the elements of set B is present in set A
⟹Set B is subset of set A

### Set A & B have common elements

In the above figure;

(a) Rectangle U represents universal set.

(b) Circle A represents set A.

(c) Circle B represents set B

(d) The red area represents common element between set A & B

### Disjoint set A & B with no common elements

In the above figure;

⟹ Rectangle U represents universal set

⟹ Circle A & B represents set A & B.

Both the circles are away from each other as there are no common elements between the two sets.

## Representing set operation using Venn diagram

Set operation can also be represented by Venn diagram.

Here we will discuss following operations;

(a) Union of Sets
(b) Intersection of sets
(c) Difference of sets
(d) Complement of sets

### Union of sets using Venn diagram

Suppose A & B are given two sets.

The union of set A & B will result in all elements present in set A & B.

In Venn diagram, Union of set is represented as;

In the above figure;

⟹ Rectangle U represents universal set

⟹ Circle A & B represents set A & B respectively.

⟹ The area colored in blue represents union of set A & B.

### Intersection of Sets using Venn diagram

Let A & B are two given sets.

The intersection of set A & B results in common element present in A & B both.

Venn diagram representation of set intersection is given below.

In the above figure;

⟹ Rectangle U represents universal set.

⟹ Circle A & B represent set A & B respectively.

⟹ the green area represent the intersection of set A & B.

### Difference of sets using Venn diagram

Let A & B are two given sets.

The difference of set A – B will result in removal of common elements of set A & B from set A.

In Venn diagram, the difference is shown as follows;

In the above figure;

⟹ Rectangle U represents Universal set.

⟹ Circle A & B represents set A & B respectively.

⟹ From circle A, the common elements of A & B (shown by green area) is removed.

Hence, the blue part of the area represent the subtraction A – B.

### Complement of Set

Let A be the given set.

The complement of set A contain all the elements which are not in A

In Venn Diagram, the complement is represented as follows;

In the above figure, the green area represents the complement of set A.

## Other notable examples of Venn diagram representation

(a) \mathtt{( \ A\ \cup \ B\ )^{'}}

Given above is the complement of A union B.

Here we will draw Venn diagram in two steps;

(i) Venn diagram of Union of set A ∪ B

The green area represents A ∪ B

(ii) Now take the complement of A ∪ B

Here the green area represents \mathtt{( \ A\ \cup \ B\ )^{'}} .

Hence the above image is the right representation.

(ii) Represent \mathtt{( \ A\ \cap \ B\ )^{'}} in Venn diagram

Given above is the complement of A intersection B.

We will draw the Venn diagram in two steps;

(a) First Draw Venn diagram of A ∩ B

The blue area represents the expression A ∩ B .

(b) Now take the complement of A ∩ B

The area in blue represents the solution.

(iii) Represent \mathtt{( \ A\ -\ B\ )^{'}} using Venn diagram

Given above is the complement of A difference with B.

We will complete the Venn diagram in two steps;

(a) First draw Venn diagram of A – B

The area with green color represents A – B.

(b) Now take the complement of A – B

The area with green color represents the solution.