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
  
  \(A∩A′ = ∅\)
  
  Law of double complementation: The complement of the complement of a set returns the original set.
  
  \((A′)′ = A\)
   
• Law of empty set and universal set: This law defines the complements of both 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'.

Here are some parent and learner friendly tips and tricks for beginners to master complement of a set:
 

1. Always identify the universal set first. The universal set is the foundation, and it defines the "universe of elements" we are working with. We cannot find A' without U.
    

2. Use Venn diagrams to visualize. Draw circles for sets inside a rectangle representing 𝑈. The shaded part outside 𝐴 (but inside 𝑈) represents 𝐴′. Visualization helps make the concept concrete and prevents errors.
    

3. Use De Morgan’s laws to simplify complex problems. When faced with multiple sets: Replace ∪ with ∩ and ∩ with ∪, Then complement each individual set.
   
   Example: \((A∪B∪C)′=A′∩B′∩C′\)
    

4. Relate to everyday life. If \(U = all \ fruits\), and \(A = tropical \ fruits\), then \(A′ = non-tropical\ fruits\). Using such relatable examples makes it easier to remember.
    

5. Test yourself with quick quizzes. Try writing small universes (like 1–10) and sets, and practice finding complements. Keep track using Venn diagrams or count formulas to cross-check your results.

The complement of a set concept is widely applicable across various fields, aiding in decision-making and analysis. Some of the applications of the complement of a set are:  
 

• 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. Then the complement of B would include bananas, oranges, and grapes.
   
• 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, such as books about space, history, or fairy tales.
   
• Birthday invitations: You are 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."