Complement of a Set
2026-02-28 17:14 Diff
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Last updated on December 10, 2025

The complement of a set is made up of all elements that are not present in the set, but present within a larger context known as the universal set. The complement of a set B is denoted by B′.

What is a Set?

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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.

What is the Complement of a Set?

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.

What are the Properties of the Complement of a Set?

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

    \((A∩B)′ = A′∪B′\)

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What is the Symbol for the Complement of a Set?

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.

Venn Diagram of the Complement of a Set

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 (∅).

How to Find the Complement of a 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'.

Tips and Tricks to Master Complement of a Set

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.

Common Mistakes in Complement of a Set and How to Avoid Them

Usually, the complement of a set is a concept of set theory. In the beginning, it can be confusing to the students. Identifying these mistakes and learning how to avoid them is important for accurate mathematical reasoning.

Real-Life Applications of the Complement of a Set.

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."

Problem 1

Given the universal set U = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} and set B = {Monday, Tuesday, Wednesday, Thursday}, find B'.

Okay, lets begin

\(B' = \{Friday, Saturday, Sunday\}\)

Explanation

The complement of B consists of days in U not in B.

\(U = \{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday\}\)

set \(B = \{Monday, Tuesday, Wednesday, Thursday\}\)

\(B' = U - B \\ B' = \{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday\} - \{Monday, Tuesday, Wednesday, Thursday\}\\ B' = \{Friday, Saturday, Sunday\}\)

Well explained 👍

Problem 2

Let U = {1, 2, 3, 4, 5, 6, 7} and A = {1, 2, 3, 4}, find A'.

Okay, lets begin

\(A' = \{5, 6, 7\}\)

Explanation

The complement of A consists of elements in U that are not in A.

\(U = \{1, 2, 3, 4, 5, 6, 7\}\)

\(A = \{1, 2, 3, 4\}\)

\(A' = U - A\\ A' = \{1, 2, 3, 4, 5, 6, 7\} - \{1, 2, 3, 4\}\\ A' = \{5, 6, 7\}\)

Well explained 👍

Problem 3

If U = {a, b, c, d, e, f, g} and C = {a, c, e}, find C'.

Okay, lets begin

\(C' = \{b, d, f, g\}\)

Explanation

C' includes all elements in U that are not in C

\(U = \{a, b, c, d, e, f, g\}\)

\(C = \{a, c, e\}\)

\(C' = U - C\\ C' = \{a, b, c, d, e, f, g\} - \{a, c, e\}\\ C' = \{b, d, f, g\}\)

Well explained 👍

Problem 4

Given U = {apple, banana, orange, pear, mango} and D = {apple, banana, orange}, find D'.

Okay, lets begin

\(D' = \{pear, mango\}\)

Explanation

D' contains fruits in U that are not in D.

\(U = \{apple, banana, orange, pear, mango\}\)

\(D = \{apple, banana, orange\}\)

\(D' = U - D\\ D' = \{apple, banana, orange, pear, mango\} - \{apple, banana, orange\}\\ D' = \{pear, mango\}\)

Well explained 👍

Problem 5

Let U = {1, 2, 3, 4, 5, 6} and E = {2, 4, 6}, find E'.

Okay, lets begin

\(E' = \{1, 3, 5\}\)

Explanation

E' includes elements in U not in E.

\(U = \{1, 2, 3, 4, 5, 6\}\)

\(E = \{2, 4, 6\}\)

\(E' = U - E\\ E' = \{1, 2, 3, 4, 5, 6\} - \{2, 4, 6\}\\ E' = \{1, 3, 5\}\)

Well explained 👍

FAQs of the Complement of a Set

1.What is De Morgan's Law of complements?

De Morgan's Law states the complement of the union of two sets as the intersection of their complements: (A ∪ B)' = A' ∩ B'. The complement of the intersection of two sets will be the union of their complements: (A ∩ B)' = A' ∪ B'. These laws help students simplify expressions involving complements

2.How do we find the complement of a set?

To find the complement of a set, make a list of all the elements in the universal set and exclude the elements of the given set. The elements that remain form the complement.

3.What is the complement of a set in a Venn diagram?

In a Venn diagram, the complement of a set is defined by shading the area outside the circle defining the set, inside the rectangle defining the universal set.

4.What is the complement of a set?

The complement of a set includes all the elements in the universal set which are not in the given set. For instance, if the universal set is {1, 2, 3, 4, 5, 6, 7} and the set A is {1, 3, 7}, then the complement of A will be {2, 4, 5, 6}. 

5.How do we denote the complement of a set?

A is denoted as A' in the complement of a set. For instance, if U is a set of integers, and A is a set of even numbers, then its complement A' is going to be a set of odd numbers.

6.How can I make this concept easier for my child to understand?

Use real-life examples. Let U = all fruits you have at home. Let A = apples and oranges. Then A′ = all the other fruits (like bananas, grapes, etc.) that are not apples or oranges. You can also use toys, books, or colored blocks to make it visual.

7.How do I help my child visualize the complement of a set?

Use Venn diagrams. Draw a rectangle for the universal set and a circle for the subset A. Shade the region outside the circle but inside the rectangle — that’s the complement A′. Children grasp this idea quickly through visuals.

8.My child often gets confused between complement and subset. How can I clarify the difference?

Tell them the definition clearly. A subset means “part of” another set. A complement means “everything outside” a set (but still in the universal set). Use visuals or simple words like “inside vs. outside” to make it clearer.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.