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 neither A ⊂ B nor B ⊂ A
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 neither A ⊂ B nor B ⊂ A
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 neither A ⊂ B nor B ⊂ A
A – B when A and B are disjoint sets.
Here A – B = A
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)
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 are disjoint.
Here, we observe that there is no common element in A and B.
Therefore, n(A ∪ B) = n(A) + n(B)
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)
A – B
B – A
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
● Finite Sets and Infinite Sets
● Problems on Intersection of Sets
● Problems on Complement of a Set
● Problems on Operation 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
8th Grade Math Practice
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