Number of elements in sets n(A) and Some Results on Them

Note union operation and universal set have the same symbol in these pages. Hence identify appropriately.

n(A) denotes the number of elements in the set A. Similarly n(B) and n(C).

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)