Complement of a Set

The complement of a set is the set that includes all the elements of the universal set that are not present in the given set. Let's say A is a set of all coins which is a subset of a universal set that contains all coins and notes, so the complement of set A is a set of notes (which does not includes coins).

In this article we will discuss in detail the complement of a set, its definition along with properties, solved examples, and practice questions.

If universal set (U) is having a subset A then the complement of a set A which is represented as A' contains the elements other than the elements of set A. i.e., A' includes the elements of the universal set but not the elements of set A. Mathematically the complement of a set A is written as, A' = {x ∈ U : x ∉ A}. In other words, the complement of a set A is the difference between the universal set and set A. i.e., A' = U - A.

Complement of Set Symbol

The complement of any set is is represented as A', B', C' etc. In other words, we can say, if the universal set is (U) and the subset of the universal set (A) is given then the difference between universal set (U) and the subset of the universal set (A) is the complement of the subset, that is

• A' = U - A [OR]
• A' = {x ∈ U : x ∉ A}

Example of Complement of a Set

The procedure of finding the complement of a set is demonstrated by an example here. If the universal set is all prime numbers up to 25 and set A = {2, 3, 5} then the complement of set A contains elements other than the elements of A.

• Step 1: Check for the universal set and the set for which you need to find the complement. U = {2, 3, 5, 7, 11, 13, 17, 19, 23}, A = {2, 3, 5}.
• Step 2: Subtract, that is (U - A). Here,
  U - A = A'
  = { ̶2̶,̶ 3̶,̶ 5̶, 7, 11, 13, 17, 19, 23} - { ̶2̶,̶ 3̶,̶ 5̶}
  = {7, 11, 13, 17, 19, 23}

For a better understanding look at the complement of a set Venn diagram given below which clearly shows the complement of set A that is A'. Here A' is not part of set A and set A is also not a part of A'. i.e., A and A' are two disjoint sets. Also, A and A' are subsets of U.

• Here, the shaded portion in orange shows the set A
• The shaded portion in white shows the complement of set A (A').

Following are the properties of the complement of a set that includes complement laws, the law of double complementation, the law of empty set and universal set, and de Morgan's law.

Complement Laws

• If A is a subset of the universal set then A' is also a subset of the universal set, therefore the union of A and A' is the universal set, represented as A ∪ A’ = U
• The intersection of Set A and A' provides the empty set “∅”, represented as A ∩ A’ = ∅

For example, If U = {1, 2, 3, 4, 5} and A = {4 , 5}, then A' = {1, 2, 3 }. Now, notice that A ∪ A’ = U = {1, 2, 3, 4, 5}. Also, A ∩ A’ = ∅

Law of Double Complementation

• In this law, the complement of the complemented set is the original set, (A')' = A
• The complement of the set A′, where A′ itself is the complement of A, the double complement of A is thus A itself.

In earlier example, U = {1, 2, 3, 4, 5} and A = {4 , 5} then A' = {1 , 2 , 3 }.
The complement of A' = (A')' = {4, 5}, which is equal to set A.

Law of Empty set and Universal Set

• The complement of the universal set is an empty set or null set (∅) and the complement of the empty set is the universal set.
• Since the universal set contains all elements and the empty set contains no elements, therefore, their complement is just opposite to each other, represented as ∅' = U And U' = ∅

In the above-given example set U = {1, 2, 3, 4, 5}, we can observe that U' = ∅ (empty set) and ∅' = {1, 2, 3, 4, 5}.

De Morgan’s law

Here are the De Morgan's laws that talk about the complement.

• The complement of the union of two sets is equal to the complement of sets and their intersection. (A U B)’ = A’ ∩ B’ (De Morgan’s Law of Union).
• The complement of the intersection of two sets is equal to the complement of sets and their union. (A ∩ B)’ = A’ U B’ (De Morgan’s Law of Intersection).

Here is an example for proving De Morgan's law, U = {1, 2, 3, 4, 5} and A = {4, 5} and B = {1, 2}. Thus,

De Morgan’s Law of Union: (A U B) = {1, 2, 4, 5} and (A U B)' = {3} and thus, A' ∩ B' = {1, 2, 3} ∩ {3, 4, 5} = {3}. Thus, so (A U B)’ = A’ ∩ B’ = {3}.

De Morgan’s Law of Intersection: (A ∩ B) = ∅ (empty), (A ∩ B)' = {1, 2, 3, 4, 5} and thus, A' U B' = {1, 2, 3 } U {3, 4, 5} = {1, 2, 3, 4, 5}. Thus, (A ∩ B)’ = A' U B'

Important Notes on the Complement of a Set:

• The complement of a set A is denoted by A' and is obtained by subtracting A from the universal set U. i.e., A' = U - A.
• A set and its complement are always disjoint.
• The complement of the universal set is the empty set or null set.

☛ Related Topics:

• Sets Formula
• Venn Diagram Formula
• Set Builder Notation

What is the Complement of a Set?

The complement of set A is defined as a set that contains the elements present in the universal set but not in set A. For example, Set U = {2, 4, 6, 8, 10, 12} and set A = {4, 6, 8}, then the complement of set A, A′ = {2, 10, 12}.

How To Find the Complement of a Set?

If the universal set(U) is given and another set A which contains some elements of the universal set is given, we can find the complement of the set A, represent as A'. The elements that are not part of set A but part of set U will be the elements of A', which is the complement of set A. Here, A' = {x ∈ U: x ∉ A}.

What is the Complement of the Universal Set?

The universal set contains all the possible elements whereas the null set contains no elements at all. Thus, a complement of the universal set is the null set.

What is the Complement of a Set A if Universal Set is the Set of Letters in the English Alphabet and Set A is the Set of Consonants in the English Alphabet?

If the universal set is all alphabets and set A contains all the consonants then the complement of set A, that is A' will be the set of vowels of the English alphabets.

What is A Intersection B Complement?

A intersection B complement is nothing but A complement union B complement. (A ∩ B)’ = A’ U B’. This is known as the De Morgan's law of intersection.

What is the Complement of the Intersection of Sets?

The complement of the intersection of sets is the set of elements that are included in the universal set U but not included in the intersection set. For example, suppose set U = set of natural numbers less than 10 and elements of set X = {1, 2, 5, 6} and set Y = {1, 3, 4, 5, 6}. Thus, the intersection of set X and Y or X ∩ Y = {1, 5, 6} and complement of (X ∩ Y) or (X ∩ Y)' = {2, 3, 4, 7, 8, 9}.

One can easily prove that the complement of the intersection of two sets is equal to the union of the complement of each of the sets.

What is the Complement of an Empty Set or Null Set?

An empty set means there are no elements in the set, so the complement of an empty set or a null set is the universal set containing all the elements.

What is A Union B Complement?

A union B complement is nothing but A complement intersection B complement. (A U B)’ = A’ ∩ B’. This is known as the De Morgan's law of union of sets.