Complement of a Set

A set is a collection of specific objects. For example, the collection of coins will be one set, and stickers will be another set. Consider a set that has all universal elements. The complement of a set means that set in which these universal elements are absent. For example, in a collection of coins, notes will not be there. Therefore, the collection of notes is a complementary set of collections of coins.

Let us consider another example. A universal set contains all prime numbers up to 25, and set A has elements 2, 3, 5. We can represent the complement of the set A as:

1. Check the elements of the universal set for which we need to find the complement. Here, U = {2, 3, 5, 7, 11, 13, 17, 19, 23}, A = {2, 3, 5}.
2. Find the difference between the set A and the universal set by eliminating the elements: U - A = A' = {2, 3, 5, 7, 11, 13, 17, 19, 23} - {2, 3, 5}.
3. Therefore, the complement of set A = A' = {7, 11, 13, 17, 19, 23}

Complement of a set: Definition

The complement of a set will not have elements of the main set. These elements will be present in the universe but not in the complement of a set. If a universal set has an A subset, then the complement of A is denoted by A’. Mathematically, we represent it as: A' = {x ∈ U : x ∉ A}

In addition, the complement of a set is the difference between a set A and the universal set.

The Venn diagram for complement of a set is represented as:

Properties of complement of a set

  1) Complement laws
If A is a set of the universal set U, then the complement of the set, A’, will also be a set of the universal set.
A U A’ = U
The intersection of A and A’ = 0 or null, denoted by ∅.
A ∩ A’ = ∅

For example, if U = 2, 3, 4, 5, 6, 7, 8
A = 3, 4, 5
A’ = 2, 6, 7, 8
Then A U A’ = U = 2, 3, 4, 5, 6, 7, 8
Also, A ∩ A’ = ∅, since no terms are common between A and A’.   

2) Law of double complementation
The complement of the complement set is the original set A.
(A’)’ = A
If U = 2, 3, 4, 5, 6, 7, 8
A = 3, 4, 5
A’ = 2, 6, 7, 8
(A’)’ = A = 3, 4, 5   

3) Law of empty set
and universal set The complement of a universal set is an empty set or null set. Since the universe contains all the elements, therefore, the complement of that set will be null, i.e., no elements will be present. ∅' = U and U' = ∅   

4) De Morgan’s law of complement of a set
The complement of union of two sets is equal to the intersection of the individual sets.
(A U B)’ = A’ ∩ B’
The complement of intersection of two sets is equal to the union of their complements.
(A ∩ B)’ = A’ U B’