# Theorems on Cardinal Numbers

Let A, B, C are finite sets in a finite universal set U. Then

- n (A âˆªB) = n(A) + n(B) â€“ n (A âˆ© B)
- n (A âˆª B) = n(A) + n (B) A and B are disjoint non void sets.
- n (A âˆª B âˆª C) = n(A) + n(B) + n(C) â€“ n(A âˆ© B) â€“ n(B âˆ© C) â€“ n(C âˆ© A) + n(A âˆ© B âˆ© C)
- n (Aâ€“B) = n (A) â€“ n (A âˆ©B) = n (A âˆ© B')
- n (A âˆ† B) = n (A) + n (B) â€“ 2 n (A âˆ© B)
- n(A') = n (U) â€“ n (A)
- n (A' âˆª B') = n (U) â€“ n (A âˆ© B)
- n (A' âˆ© B') = n (U) â€“ n (A âˆª B)
- If A
_{1}, A_{2}, ......, A_{n}are disjoint sets, then

**De Morganâ€™s laws**

- (A âˆª B)' = A'âˆ© B'
- (A âˆ© B)' = A' âˆª B'
- A â€“ (B âˆª C) = (A â€“ B) âˆ© (A â€“ C)
- A â€“ (B âˆ© C) = (A â€“ B) âˆª (A â€“ C)