If set Q is a subset of the universal set ∪, then the complement of the set Q is the set of objects or members which belongs to the universal set ∪ but do not belong to the set Q. The universal set is a set that contains all elements in other sets and all other elements in discussions. The complement of the set is often denoted as Q’. i.e set Q is relative to the universal set.

## What is a complement of a set?

Complement of a set refers to sets of objects or elements that are contained in the universal set but are not in the given set. For instance, if set A is a subset of the universal set, then the complement of A are elements that are found in the universal set but they are not in set A. i.e A’ is relative to the universal set.

## How to find the complement of a set

In general, the complement of a given set is equal to the members or objects in the universal set minus elements or objects in the given set. The examples below show a clear explanation of the complement of the set. Let’s look at the example below

Example 1: if the universal set U ={1,2,3,4,5,6,7,8,9,10} set B is a subset of the universal set {6,8,10}. Then the complement of set B i.e B’ ={1,2,3,4,5,7,9}.

Example 2: if set Q and P are subsets of the universal set U={4,5,6,7,8,9,10,11,12,13,14,15,16,17} and set Q={even numbers} and set P={odd numbers}. List the members of the set Q, P, Q’, P’, (Q U P)’.

solutions

a) Q={4,6, 8,10,12,14,16,}

b) P={5,7,9,11,13,15,17}

c) Q’={5,7,9,11,13,15,17}

d) P’={4,6,8,10,12,14,16}

e) (Q U P)’ ={ } or ∅

## Complement of a set in a Venn diagram

The set is represented by circles inside a rectangle representing the universal set. The complement of the set is the region outside the circle as shown in the Venn diagram below.