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Theorems on Cardinal Numbers

Let A, B, C are finite sets in a finite universal set U. Then
  1. n (A B) = n(A) + n(B) – n (A B)
  2. n (A B) = n(A) + n (B) Description: 781.pngA and B are disjoint non void sets.
  3. n (A B C) = n(A) + n(B) + n(C) – n(A B) – n(B C) – n(C A) + n(A B C)
     
    The results (i) and (iii) can be extended to any number of sets.
  4. n (A–B) = n (A) – n (A B) = n (A B')
  5. n (A ∆ B) = n (A) + n (B) – 2 n (A B)
  6. n(A') = n (U) – n (A)
  7. n (A' B') = n (U) – n (A B)
  8. n (A' B') = n (U) – n (A B)
  9. If A1, A2, ......, An are disjoint sets, then
     
    Description: 786.png 
De Morgan’s laws
  1. (A B)' = A'B'
  2. (A B)' = A' B'
  3. A – (B C) = (A – B) (A – C)
  4. A – (B C) = (A – B) (A – C)




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