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Complement of a Set

The complement of a set A (also called “absolute complement” of A) is the set of all those elements of the universal set S which are not elements of A. It is denoted by Ac or A′. Clearly Ac or A′ = S - A.
 
Symbolically, Ac or A′ = {x : x S and x A}. Thus x Ac x A.
 
Complement of a set can be represented by Venn diagram as shown in Figure. The shaded region represents A′.
 
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Important properties of complement

  1. A A′ = φ
  2. A A′ = U
  3. U′ = φ
  4. (A′)′ = A
  5. A B B A
  6. A - B = B′ - A
  7. A - B = A B
  8. B - A = A B

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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