Properties of Elements in Sets

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 non-void 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

Representation of a Set

Types of Sets

Pairs of Sets

Subset

Practice Test on Sets and Subsets

Complement of a Set

Problems on Operation on Sets

Operations on Sets

Practice Test on Operations on Sets

Word Problems on Sets

Venn Diagrams

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Examples on Venn Diagram

Practice Test on Venn Diagrams

Cardinal Properties of Sets



7th Grade Math Problems

8th Grade Math Practice

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