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
From Properties of Elements in Sets to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.