Complement of a Set

A set is a collection of objects referred to as elements. These elements are grouped because they share a common attribute or because specific rules specify the set. We denote sets using curly brackets “{ }” and separate each element with a comma.

For example, if we list the first few whole numbers, we can express the set as W = {0, 1, 2, 3, 4, 5, …}, where W represents the set of whole numbers starting from 0.

The complement of a set A is the collection of all elements in the universal set U that are excluded from A.

It’s written as A′ = {x ∈ U | x ∉ A} or A′ = U ∖ A. For example, if U = {1, 2, 3, 4, 5} and A = {2, 4}, then A′ = {1, 3, 5}, consisting of elements excluded from A.
 

Understanding the properties of the complement of a set will help in solving problems related to intersections, unions, and set relationships.

Below are some of its properties:

Complement Laws

These laws define the relationship between a set and its complement within a universal set:
Union Law: The union of set A and its complement is identical to the universal set U
 A∪A′ = U 

Intersection Law

The intersection of a set A and its complement A′ is the empty set
 A∩A′ = ∅ 
2. Law of Double Complementation
Taking the complement of the complement of a set returns the original set:
 (A′)′ = A 

Law of Empty Set and Universal Set

This defines the complements of the empty set and the universal set:
The complement of the empty set is the universal set:
 ∅′ = U
The complement of the universal set is defined as the empty set:
 U′ = ∅ 

De Morgan’s Laws

These laws relate to the complement of unions and intersections of sets:

• First Law: The complement of the union of two sets is the intersection of their complements:

                (A∪B)′ = A′∩B′ 

• Second Law: The complement of the intersection of two sets is the union of their complements:

                (A∩B)′ = A′∪B′ 
 

The complement of a set A, denoted as A′, includes all the elements in the universal set U that are not in A.

This is expressed as:
A′ = {x ∈ U ∣ x ∉ A}

We can also write this as:
A′ = U - A 

This means that A′ includes every element of U except those that are in A.
 

In the given Venn diagram, the universal set U holds two subsets: A and A'.

The green-shaded area is defined as set A, the red-shaded area is defined as set A'.

As A and A' have no elements in common, they are separate sets and their intersection is the empty set (∅).
 

The complement of a set can be found by excluding the elements of the given set from the universal set.

Example
Universal Set (U): {1, 2, 3, 4, 5, 6, 7}

Given Set (A): {1, 3, 7}

Complement of A (A'):
A' = U - A = {1, 2, 3, 4, 5, 6, 7} − {1, 3, 7}

A' = {2, 4, 5, 6}
Step 1. Identify the Universal Set (U): Define the set that includes all possible elements.

Step 2. Define the Given Set (A): Identify the set for which you want to find the complement.

Step 3. Subtract Elements of A from U: List all elements in U that are not in A.

Step 4. Express the Complement: The result is the complement of A, denoted as A'.

• Classroom Attendance: All enrolled students in the school are represented by the universal set U. If we define a set A as all students who are present today, then the complement of A (A') would be the absent students. 

• Fruit Basket: There is a basket having different kinds of fruits. If we define set B as the apples in the basket, then the complement of B would be all the fruits in the basket that are not apples. So if there are bananas, oranges, and grapes, they will be in set B.

• Toy Box: Imagine there is a toy box filled with different toys. If set C represents the robots in the box, then the complement of C (C') would include all the toys that are not robots like dolls, cars, or puzzles.

• Library Books: In a library, there are many books. If set D represents all the books about Earth, then the complement of D (D') would be all the books that are not about Earth, like books about space, history, or fairy tales.

• Birthday Invitations: You're planning a birthday party, and you invite some friends. If set E represents the friends you've invited, then the complement of E (E') would be the friends you haven't invited. It's like saying, "Everyone except those on the invitation list."