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Last updated on 19 September 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. For example, the universal set (U) is all students in a school. In subset (B) are students who play a musical instrument. Now, the complement of B (denoted by B′ or U - B) is students who do not play any musical instrument.
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:
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.
The intersection of a set A and its complement A′ is the empty set
A∩A′ = ∅
2. Law of Double Complementation
The complement of the complement of a set returns the original set.
(A′)′ = A.
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′ = ∅.
These laws relate to the complement of unions and intersections of sets.
(A∪B)′ = A′∩B′
(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 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'.
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.
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:
Given the universal set U = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} and set B = {Monday, Tuesday, Wednesday, Thursday}, find B'.
B' = {Friday, Saturday, Sunday}
The complement of B consists of days in U not in B.
Let U = {1, 2, 3, 4, 5, 6, 7} and A = {1, 2, 3, 4}, find A'.
A' = {5, 6, 7}
The complement of A consists of elements in U that are not in A.
If U = {a, b, c, d, e, f, g} and C = {a, c, e}, find C'.
C' = {b, d, f, g}
C' includes all elements in U that are not in C
Given U = {apple, banana, orange, pear, mango} and D = {apple, banana, orange}, find D'.
D' = {pear, mango}
D' contains fruits in U that are not in D.
Let U = {1, 2, 3, 4, 5, 6} and E = {2, 4, 6}, find E'.
E' = {1, 3, 5}
E' includes elements in U not in E.
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.
: He loves to play the quiz with kids through algebra to make kids love it.