# Complement of a Set using Venn Diagram

The complement of a set using Venn diagram is a subset of U. Let U be the universal set and let A be a set such that A ⊂ U. Then, the complement of A with respect to U is denoted by A' or A$$^{C}$$ or U – A or ~ A and is defined the set of all those elements of U which are not in A.

Thus, A' = {x ∈ U : x ∉ A}.

Clearly, x ∈ A' ⇒ x ∉ A

(A – B) is also called the complement of B relative to A. From the definition it is clear that the complement of the whole set in a set is the null set; for U' = U – U = ∅ again ∅' = U - ∅ = U also (A')' = U – A' = U – (U – A) = A. If the set of real numbers be the universal set, then the set of rational numbers and the set of irrational numbers are complements of each other.

Example on complement of a set using Venn diagram:

1. Let the set of natural numbers N = {1, 2, 3, ………..} be the universal set and let A = {2, 4, 6, 8, ……….}

Then A' = {1, 3, 5, ………}

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

3. If U = {1, 2, 3, 4, 5, 6} and A = {2, 3, 4} then U – A = ~ A = A' = {1, 5, 6}.

4. U = {1, 2, 3, 4, 5, 6} be the universal set and A = {1, 3, 5} then A' = {2, 4, 6}.

Properties of complement of a set:

1. U' = ∅

2. ∅' = U

3. A U A' = U        For any subset A

4. A ∩ A' = ∅        For any subset A

5. (A')' = A          For any subset A.

Set Theory

Sets