Complement of a Set

Before studying about the Complement of a set, let us understand what are sets?

Sets Definition

A well-defined collection of objects or elements is known as a set. Any set consisting of all the objects or elements related to a particular context is defined as a universal set. It is represented by [latex]U[/latex].
For any set A which is a subset of the universal set [latex]U[/latex], the complement of the set [latex]A[/latex] consists of those elements which are the members or elements of the universal set [latex]U[/latex] but not of the set [latex]A[/latex]. The complement of any set [latex]A[/latex] is denoted by [latex]A'[/latex].

Also, read:

Complement of a Set Definition

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

[latex]A'[/latex] = {[latex]{x ~:~ x~ ∈ ~U ~and ~x ~∉ ~A}[/latex]}

Alternatively it can be said that the difference of the universal set [latex]U[/latex] and the subset [latex]A[/latex] gives us the complement of set [latex]A[/latex].

Venn Diagram for the Complement of a set

The Venn diagram to represent the complement of a set A is given by:

complement of a set

Complement of a Set Examples

To make it more clear consider a universal set [latex]U[/latex] of all natural numbers less than or equal to 20.

Let the set [latex]A[/latex] which is a subset of [latex]U[/latex] be defined as the set which consists of all the prime numbers.

Thus we can see that [latex]A[/latex] = {[latex]{2, 3, 5, 7, 11, 13, 17, 19}[/latex]}

Now the complement of this set A consists of all those elements which is present in the universal set but not in [latex]A[/latex]. Therefore, [latex]A'[/latex] is given by:

[latex]A'[/latex]={[latex]{1,4,6,8,9,10,12,14,15,16,18,20}[/latex]}

Example: Let [latex]U[/latex] be the universal set which consists of all the integers greater than 5 but less than or equal to 25. Let [latex]A[/latex] and [latex]B[/latex] be the subsets of [latex]U[/latex] defined as:

[latex]A[/latex] = {[latex]{x~:x~ ∈~U ~and~ x~ is~ a~ perfect~ square}[/latex]}

[latex]B[/latex] = [latex]{7, 9, 16, 18, 24}[/latex]

Find the complement of sets A and B and the intersection of both the complemented sets.

Solution: The universal set is defined as:

[latex]U[/latex] = {[latex]{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}[/latex]}

Also, [latex]A[/latex] = {[latex]{9,16,25}[/latex]} and

[latex]B[/latex] = {[latex]{7,9,16,18,24}[/latex]}

The complement of set A is defined as:

[latex]A'[/latex] = {[latex]{x~:~x~∈~U~ and ~x~∉~A}[/latex]}

Therefore, [latex]A'[/latex] = {[latex]{6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24}[/latex]}

Similarly the complement of set B can be given by:

[latex]B'[/latex] = {[latex]{6,8,10,11,12,13,14,15,17,19,20,21,22,23,25}[/latex]}

The intersection of both the complemented sets is given by [latex]A’∩ B'[/latex].

Rightarrow A’∩ B’= {6, 8, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23}

We can see from the above discussions that if a set [latex]A[/latex] is a subset of the universal set [latex]U[/latex] then the complement of set [latex]A[/latex] i.e. [latex]A'[/latex] is also a subset of [latex]U[/latex]..

Now we are clear on the concept of the complement of sets. There is a lot more to learn. Enrich your knowledge and reach new horizons of success by downloading BYJU’S – The Learning App to know more or visit our website.

 

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