Saturday, June 7, 2008

Ch. 1. Sets - Number of elements in sets

Some results on Number of elements in sets n(A), n(B), and n(C)

Note union operation and universal set have the same symbol

If A,B and C are finite sets. U is the finite universal set, then

(i) n(A U B) = n(A) +n(B) – n(A∩B)

(ii) n(A U B) = n(A) +n(B)  A, B are disjoint non-void sets.

(iii) n(A-B) = n(A) –n(A∩B)

(iv) n(A ΔB) = Number of elements which belong to exactly one of A or B
= n((A-B) U (B-A))

(v) n(A U B U C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C)+n(A∩B∩C)

(vi) No. Of elements in exactly two of the sets A,B,C
= n(A∩B) + n(B∩C)+n(C∩A)-3n(A∩B∩C)

(vii) No. 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)

(viii) n(A’ U B’) = n((A∩B)’) = n(U) – n(A∩B)

(ix) n(A’∩B’) = n((AUB)’) = n(U)-n(A∩B)

Laws of algebra of sets

1. Idempotent laws

A U B = A

A∩A = A

2. Identity laws

3. Commutative laws

4. Associative laws

5. Distributive laws

6. De-Morgan's laws

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