Complement of Sets

Let U be the universal set and A be a subset of U. Then, the complement of A is the set consisting of all elements of U which do not belong to A.
Complement of A with respect to (w.r.t) U is denoted by A' or $\displaystyle \small A^{c}$ or Ā
A'={x: x∈U and x∉A}
A'=U-A

Let U={1,2,3,4,5,6,7,8,9,10} and A={1,2,3,5,7,9}, then A'={4,6,8,10}

In general, if A is any subset of a set B, then the complement of A in B is the set consisting of all the elements of B which do not belong to A.

Properties of Complement Sets
Complement Laws:
(i) A∪A'=U
(ii) A∩A'=ϕ

De Morgan's Law:
(i) (A∪B)'=A'∩B'
(ii) (A∩B)'=A'∪B'

Law of double complementation:
(A')'=A

Laws of empty set and universal set
(i) ϕ'=U
(ii) U'=ϕ

Practical Problems on Union and Intersection of Sets
Let A and B be finite sets.
Number of elements in A and B is denoted by n(A) and n(B) respectively.

Case I: If A and B are disjoint, then there is no common element in A and B.
∴ n(A∪B) = n(A) + n(B)

Case II: If A and B are not disjoint, then there are common elements in A and B.
(i) n(A∪B) = n(A) + n(B) - n(A∩B)
(ii) n(A∪B) = n(A-B) + n(B-A)  + n(A∩B)
(iii) n(A) = n(A-B) + n(A∩B)
(iv) n(B) = n(B-A) + n(A∩B)
(v) n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(A∩C) + n(A∩B∩C)
(vi) n(A') = n(U) - n(A)