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.

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