∉ A, because is divisor of . Similarly, ∈ U but ∉ A, and ∈ U but ∉ A. Now , and are the only elements of U which do not belong to A. The set of these three prime numbers, i.e., the set { , , } is called the Complement of A with respect to U, and is denoted by SETS A ′ .
So we have A ′ = { , , }. Thus, we see that A ′ = { x : x ∈ U and x ∉ A }. This leads to the following definition. Definition Let U be the universal set and A a subset of U.
Then the complement of A is the set of all elements of U which are not the elements of A. Symbolically, we write A ′ to denote the complement of A with respect to U. Thus, A ′ = { x : x ∈ U and x ∉ A }. Obviously A ′ = U – A We note that the complement of a set A can be looked upon, alternatively, as the difference between a universal set U and the set A.
Example Let U = { , , , , , , , , , } and A = { , , , , }. Find A ′ . Solution We note that , , , , are the only elements of U which do not belong to A. Hence A ′ = { , , , , }.
Example Let U be universal set of all the students of Class XI of a coeducational school and A be the set of all girls in Class XI. Find A ′ . Solution Since A is the set of all girls, A ′ is clearly the set of all boys in the class. A Note If A is a subset of the universal set U, then its complement A ′ is also a subset of U.
Again in Example above, we have A ′ = {