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Complement of a Set

Let U be the universal set and A be a subset of U. We define the complement of A with respect to U to be the set {x : x  U and x  A}. The complement of A is denoted by A'.

When the universal set is clearly understood, we usually omit the phrase "with respect to" and simply call A' as complement of A.

 

Illustration 1
Let U be the universal set of all the triangles in a plane. Let E denote the set of all the equilateral triangles in the plane, then E' = {x : x is not a equilateral triangle}.

Illustration 2
Let U be the universal set of the English alphabet and let V = {x : x is a vowel of the English alphabet}, then V' = {x : x is a consonant of the English alphabet}.
We now note some facts about the complement of a set.
  1. De Morgan's Laws
    (A B)' = A' B'; (A B)' = A' B';
  2. Complement Laws
    U' = φ and φ ' = U
    That is, the complement of the universal set is the null set φ and vice versa.
  3. Involution Law
    (A')' = A
    That is, the complement of the complement of a set is the set itself.




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