Relationship in Sets using Venn Diagram

The relationship in sets using Venn diagram are discussed below:

The union of two sets can be represented by Venn diagrams by the shaded region, representing A ∪ B.

 A ∪ B when A ⊂ B






A ∪ B when A ⊂ B


A ∪ B when neither A ⊂ B nor B ⊂ A






A ∪ B when neither A ⊂ B nor B ⊂ A


A ∪ B when A and B are Disjoint Sets






A ∪ B when A and B are disjoint sets


The intersection of two sets can be represented by Venn diagram, with the shaded region representing A ∩ B.

A ∩ B when A ⊂ B, i.e., A ∩ B = A






A ∩ B when A ⊂ B, i.e., A ∩ B = A


A ∩ B when neither A ⊂ B nor B ⊂ A





A ∩ B when neither A ⊂ B nor B ⊂ A


A ∩ B = ϕ No shaded Part





A ∩ B = ϕ No shaded part



The difference of two sets can be represented by Venn diagrams, with the shaded region representing A - B.

A – B when B ⊂ A





A – B when B ⊂ A


A – B when neither A ⊂ B nor B ⊂ A





A – B when neither A ⊂ B nor B ⊂ A


A – B when A and B are Disjoint Sets





A – B when A and B are disjoint sets.

Here A – B = A


A – B when A ⊂ B





A – B when A ⊂ B

Here A – B = ϕ


Relationship between the three Sets using Venn Diagram

If ξ represents the universal set and A, B, C are the three subsets of the universal sets. Here, all the three sets are overlapping sets.

Let us learn to represent various operations on these sets.

A ∪ B ∪ C







A ∪ B ∪ C


A ∩ B ∩ C







A ∩ B ∩ C


A ∪ (B ∩ C)







A ∪ (B ∩ C)


A ∩ (B ∪ C)







A ∩ (B ∪ C)


Some important results on number of elements in sets and their use in practical problems.

Now, we shall learn the utility of set theory in practical problems.

If A is a finite set, then the number of elements in A is denoted by n(A).

Relationship in Sets using Venn Diagram
Let A and B be two finite sets, then two cases arise:

A and B be Two Finite Sets
Case 1:

A and B are disjoint.

Here, we observe that there is no common element in A and B.

Therefore, n(A ∪ B) = n(A) + n(B)


A and B are not Disjoint Sets

Case 2:

When A and B are not disjoint, we have from the figure

(i) n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

(ii) n(A ∪ B) = n(A - B) + n(B - A) + n(A ∩ B)

(iii) n(A) = n(A - B) + n(A ∩ B)

(iv) n(B) = n(B - A) + n(A ∩ B)


Sets A – B



A – B


Sets B – A



B – A


A ∩ B Sets



A ∩ B


Let A, B, C be any three finite sets, then

n(A ∪ B ∪ C) = n[(A ∪ B) ∪ C]

                  = n(A ∪ B) + n(C) - n[(A ∪ B) ∩ C]

                  = [n(A) + n(B) - n(A ∩ B)] + n(C) - n [(A ∩ C) ∪ (B ∩ C)]

                  = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)

                     [Since, (A ∩ C) ∩ (B ∩ C) = A ∩ B ∩ C]

Therefore, n(A ∪B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

Set Theory

Sets Theory

Representation of a Set

Types of Sets

Finite Sets and Infinite Sets

Power Set

Problems on Union of Sets

Problems on Intersection of Sets

Difference of two Sets

Complement of a Set

Problems on Complement of a Set

Problems on Operation on Sets

Word Problems on Sets

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Union of Sets using Venn Diagram

Intersection of Sets using Venn Diagram

Disjoint of Sets using Venn Diagram

Difference of Sets using Venn Diagram

Examples on Venn Diagram



8th Grade Math Practice

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