Complement of a Set

Complement of a Set is one of the important operations, we can perform on a set in set theory. Grouping numbers with similar properties together, i.e., arranging them in sets is what forms the foundation for the rest of mathematics, thus set theory holds a very important place in the study of mathematics. Having learned the importance of set theory, it is important to go through all its aspects to fully grasp its concept. In this article, we will learn about a particular aspect of set theory: complement of a set: its meaning, symbol, properties, and Venn diagram.

What is a Set?

In mathematics, a set is a collection or grouping of well-defined objects. All such objects when grouped together in a set are called elements. Sets are represented by capital letter symbols and the elements are placed together in a curly bracket {}.

For example, if W is the set of whole numbers, then W = {0, 1, 2, 3, 4, 5,….,∞}.

Learn more about the Set and its Type

Complement of a Set

The complement of a set is the set that consists of all the elements from the universal set which are not already included in the given set. In other words, the difference between the universal set and the given set represents the complement of that set. The complement of a set S is mathematically expressed as 

S’ = {x ∈ U and x ∉ S}

Where x denotes the elements in Universal Set, U but not in S.

For example, S is the set of all the Android mobiles, the subset of the universal set of all mobiles in the world. Here, the complement of set S would be the set of all iOS mobile phones(which does not consist of Android mobile phones).

Complement of a Set Definition

If U is a universal set and A be any subset of U then the complement of A is the set of all members of the universal set U which are not the elements of A.

Mathematically Complement of a set  can be expressed in the  following form:

• S’ = U – S
• S’ = {x ∈ U : x ∉ S}

Symbol of Complement of Set

The complements of sets A, B, C… are denoted as A’, B’, C’,… and so on. Excluding all the elements of a given set from the universal set gives us the complement of that set. 

Venn Diagram of Complement of Set

The following Venn diagram shows the universal set U and its two subsets- A and A’. The green-shaded potion represents set A and the red portion depicts the set A’. As is clear from the diagram, set A is not part of A’ and vice-versa. This implies that A and A’ are both disjoint sets or complements of each- other.

How to Find the Complement of a Set?

The complement of a set is simply found by excluding the elements of the given set from the universal set. This is shown in the example below.

Example: Find the complement of set S = {4, 8, 12, 16}, where the universal set is all multiples of 4 that are smaller than 50.

Solution:

Step 1: Identify the universal set (U) and the set (S) for which you want to find the complement.

Universal Set(U) = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}

Also, given: S = {4, 8, 12, 16}

Step 2: Calculate the complement of set S by excluding the elements of S from the universal set U i.e., S’ = U – S.

S’ = U – S = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48} – {4, 8, 12, 16}

Step 3: Simplify the resulting set to obtain the complement.

Thus, S’ = {20, 24, 28, 32, 36, 40, 44, 48}

Properties of Complement of a Set

There are various properties of the Complement of a Set, some of these properties are discussed as follows:

Complement Laws

• The union of set S and its complement set S’ is the universal set, i.e., S ∪ S’ = U.

Example: If U = {10, 11, 12, 13, 14} and S = {10, 11, 12}; then S’ = {13, 14}.

• The intersection of set S and its complement set S’ is the null or empty set, i.e., S ∩ S’ = ∅.

Example: If U = {10, 11, 12, 13, 14} and S = {10, 11, 12}; then S’ = {13, 14}.

Now, S ∩ S’ = {10, 11, 12} ∩ {13, 14} = ∅.

Law of Double Complementation

• The complement of the complement of a given set is the set itself, i.e., (S’)’ = S.

Example: If U = {10, 11, 12, 13, 14} and S = {10, 11, 12}; then S’ = {13, 14}.

The complement of S’ = U – S’ = {10, 11, 12} = set S.

Law of Null and Universal Sets

• The complement of the universal set is an empty set or null set and vice-versa, i.e., ∅’ = U And U’ = ∅.

Example: If U = {10, 11, 12, 13, 14}; then U’ = ∅ and ∅’ = U.

De Morgan’s laws

• The complement of the union of two sets is equal to the intersection of the complements of the two sets, i.e., (S U T)’ = S’ ∩ T’.

Example: If U = {10, 11, 12, 13, 14} and S = {13, 14} and T = {10, 11}.

Now, S ∪ T = {10, 11, 13, 14} and (S ∪ T)’ = {12}.

Thus, S’ ∩ T’ = {10, 11, 12} ∩ {12, 13, 14} = {12} = (S ∪ T)’.

• The complement of the intersection of two sets is equal to the union of the complements of the two sets, i.e., (S ∩ T)’ = S’ U T’.

Example: If U = {10, 11, 12, 13, 14} and S = {13, 14} and T = {10, 11}.

Now, S ∩ T = ∅ and (S ∩ T)’ = {10, 11, 12, 13, 14}.

Thus, S’ ∪ T’ = {10, 11, 12} ∪ {12, 13, 14} = {10, 11, 12, 13, 14} = S’ ∪ T’.

Sample Problems on Complement of a Set

Problem 1. If D = { x | x is a multiple of 10, x ∈ N }, find D’.

Solution:

Given: the universal set U = set of all natural numbers = N = {1, 2, 3, 4, 5, 6,….}

Also it is given that: D = { x | x is a multiple of 10, x ∈ N }

⇒ D = {10, 20, 30, 40, 50, 60,….}

⇒ D’ = U – D

D’ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12,…19, 21, 22,…, 29, 31,…..}.

Problem 2. If S and T are any two sets, prove that S‘ – T‘ = T – S.

Solution:

S’ – T’ = T – S is only possible when S’ – T’ ⊆ S – T.

Let x ∈ S’ – T’

⇒ x ∈ S’ and x ∉ T’

⇒ x ∉ S and x ∈ T (since, S ∩ S’ = ϕ )

⇒ x ∈ S – T

The above result would be true for all x ∈ S’ – T’

∴ S’ – T’ = T – S

Problem 3. How many elements are there in the complement of the set of all vowels in the English alphabet?

Solution:

The English alphabet consists of 26 alphabets of which 5 are vowels and 21 are consonants. The difference between the universal set(U) and the set of vowels(say, V) is the complement of the set of all vowels.

Now, n(U) = 26 and n(V) = 5.

Thus, n(V’) = n(U) – n(V)

⇒ n(V’) = 26 – 5

⇒ n(V’) = 21

Problem 4. If P and Q are two sets, then prove P ∩ (P‘ ∪ Q’) = P ∩ Q.

Solution:

LHS = P ∩ (P’ ∪ Q’)

Let us expand the LHS to get: (P ∩ P’) ∪ (P ∩ Q)

Since, P ∩ P’ = ϕ; our LHS becomes:

LHS = ϕ ∪ (P ∩ Q)

⇒ LHS = P ∩ Q

Thus, LHS = RHS

Hence Proved.

Problem 5. If U = {12, 13, 15, 17, 19} and S = {13, 17}, T = {12, 15, 17, 19}, then prove that (S ∪ T)’ = S’ ∩ T’.

Solution:

LHS = (S ∪ T)’. Let us first find S ∪ T.

S ∪ T = {12, 13, 15, 17, 19} = U

⇒ (S ∪ T)’ = U – (S ∪ T) = U – U = ϕ.

So, LHS = ϕ.

RHS = S’ ∩ T’

S’ = U – S = {12, 15, 19}

and T’ = U – T = {13}

⇒ S’ ∩ T’ = {12, 15, 19} ∩ {13} = ϕ

So, RHS = ϕ

Thus, LHS = RHS

Hence Proved.

Problem 6. If X and Y are two sets, prove that: X – (X – Y) = X ∩ Y.

Solution:

According to the De Morgan’s Laws, (A ∪ B)’ = A’ ∩ B’ and (A ∩ B) ‘ = A’ ∪ B’.

LHS = X – (X – Y)

⇒ LHS = X ∩ (X – Y)’

⇒ LHS = X ∩ (X ∩ Y’)’

⇒ LHS = X ∩ (X’ ∪ Y’)’) ((Y’)’ = Y)

⇒ LHS = X ∩ (X’ ∪ Y)

⇒ LHS = (X ∩ X’) ∪ (X ∩ Y)

⇒ LHS = ϕ ∪ (X ∩ Y)

⇒ LHS = (X ∩ Y)   [As ϕ ∪ A = A]

But, RHS = (X ∩ Y)

∴ LHS = RHS

Hence Proved.

FAQs on Complement of Set

Define Set.

In mathematics, a set is  defined as a collection or grouping of well- defined objects.  For example, N is the set of all natural numbers, then N = {1, 2, 3, 4, 5,….,∞}.

What is the Complement of a Set?

The complement of a set is the set that contains all the elements of the universal set which are not included in the given set and mathematically can be expressed for a set S as:

S’ = {x ∈ U : x ∉ S}

Write De Morgan’s Laws for the Complement of a Set.

De Morgan’s laws states:

• The complement of the union of two sets is equal to the intersection of the complements of the two sets, i.e., (S U T)’ = S’ ∩ T’.
• The complement of the intersection of two sets is equal to the union of the complements of the two sets, i.e., (S ∩ T)’ = S’ U T’.

What is the Complement of the Empty Set?

The empty set contains no elements however the set which contains all the possible elements is the universal set. Hence the complement of the empty set is the universal set.

What is Venn Diagram?

Venn Diagram is the diagram to represent set and their interaction using circles.