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**

● **Subset**

● **Practice Test on Sets and Subsets**

● **Problems on Operation on Sets**

● **Practice Test on Operations on Sets**

● **Venn Diagrams in Different Situations**

● **Relationship in Sets using Venn Diagram**

● **Practice Test on Venn Diagrams**

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