# Complement of a Set – Venn Diagram and Examples

Here you will learn what is the complement of a set definition with venn diagram and examples.

Let’s begin –

## Complement of a Set

Definition : Let U be the universal set and let A be a set such that A $$\subset$$ U. Then, the complement of A with respect to U is denoted by A’ or $$A^c$$ or U – A and is defined the set of all those elements of U which are not in A.

Thus, A’ = {x $$\in$$ U : x $$\notin$$ A}

Clearly,  x $$\in$$ A’  $$\iff$$  x $$\in$$ A.

## Venn Diagram :

Also Read : Formulas and Operation of Sets

Example 1 : Let the set of natural numbers N = {1, 2, 3, 4, ….. } be the universal set and let A = {2, 4, 6, 8, ….}. Then A’ = {1, 3, 5, ….}/

Example 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}.

Following results are direct consequence of the definition of the complement of a set.

(i)  U’ = {x $$\in$$ U : x $$\notin$$ U} = $$\phi$$

(ii)  $${\phi}’$$ = {x $$\in$$ U : x $$\notin$$ $$\phi$$} = U

(iii)  (A’)’ = {x $$\in$$ U : x $$\notin$$ A’} = {x $$\in$$ U : x $$\in$$ A} = A

(iv)  A $$\cup$$ A’ = U

(v)  A $$\cap$$ A’ = $$\phi$$