The following properties of elements in sets are discussed here.
If U be the universal set and A, B and C are any three finite sets then;
1. If A and B are any two finite sets then n(A  B) = n(A) – n(A ∩ B) i.e. n(A – B) + n(A ∩ B) = n(A)
2. If A and B are any two finite sets then n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
3. If A and B are any two finite sets then n(A ∪ B) = n(A) + n(B) ⇔ A, B are disjoint nonvoid sets.
4. If A and B are any two finite sets then n(A ∆ B) = Number of elements which belongs to exactly one of A or B
= n((A – B) ∪ (B – A))
= (A – B) + n(B – A) [Since (A  B) and (B – A) are disjoint.]
= n(A) – n(A ∩ B) + n(B) – n(A ∩ B)
= n(A) + n(B) – 2n(A ∩ B)
Some more properties of elements in sets using three finite sets:
5. If A, B and C are any three finite sets then n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A – C) + n(A ∩ B ∩ C)
6. If A, B and C are any three finite sets then Number of elements in exactly one of the sets A, B, C = n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C) – 2n(A – C) + 3n(A ∩ B ∩ C)
7. If A, B and C are any three finite sets then Number of elements in exactly two of the sets A, B, C = n(A ∩ B) + n(B ∩ C) + n(C ∩ A) – 3n(A ∩ B ∩ C)
8. If U be the universal set and A and B are any two finite sets then n(A' ∩ B') = n((A ∪ B)') = n(U)  n(A ∪ B)
9. If U be the universal set and A and B are any two finite sets then n(A' ∪ B') = n((A ∩ B)') = n(U)  n(A ∩ B)
● Set Theory
● Sets
● Subset
● Practice Test on Sets and Subsets
● Problems on Operation on Sets
● Practice Test on Operations on Sets
● Venn Diagrams in Different Situations
● Relationship in Sets using Venn Diagram
● Practice Test on Venn Diagrams
8th Grade Math Practice
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