Complement of a Set
Set
A set is a group of welldefined objects which have some common properties( not mandatory).

The set is denoted by capital alphabets

The elements included in the set are mainly represented by small letters

Record all the elements

Separate the elements with a comma

Encircle them in curly brackets.
Example:
X= { 2,4,6,8,10}
Subset
A group of all the elements is known as a subset of all the elements of the set is included inside another set.
Set M is said to be a subset of set N if all the elements of set M are also included in set N.
Example: If set M includes { X, Y} and set N includes { X, Y, Z) then M is the subset of N because elements of M are also included in set N.
Subset is denoted by symbol ⊆ and read as ‘is a subset of ‘
For example “ M⊆N” which means set M is a subset of set N.
Complement of a Set Definition
If U is represented as a universal set and M be any subset of the universal set (U) then the complement of set M is the set of all the elements of the U which are not the elements of set M.
M’ = { x : x ϵ U and x ∉ M}
Alternatively, it can be defined that the difference of universal set (U) and the subset M provide us the complement of set M.
Complement of a Set Examples
Consider a universal set U of all natural numbers less than or equals to 20.
Let the set M which is a subset of the universal set (U) be defined as the set which includes all the prime numbers.
Hence, we can see that M = { 2,3,5,7,11,13,17,19}
Now, the complement of this set M includes all the elements which are included in the universal set (U) but not in M,
Hence, M’ Is given by:
M’= { 1, 4,6,8, 9, 10, 12, 14, 15, 16, 18, 20}
Venn diagram for a complement of a set
The Venn diagram to represent the complement of a set M is derived by:
(image will be uploaded soon)
How do you Find the Complement of a Set?
Let us learn how to find the complement of a set through an example,
Suppose a number is randomly picked from the whole number 1 to 10. Let X be the event that number is even and less than 8. Find the complement of set X.
Steps to Find the Complement of a Set

First, separate all the numbers which are even and less than 8.

The numbers which are even and less than 8 are 2, 4, and 6.

Accordingly, the set X will be { 2, 4, 6}
Set X ={ 2,4,6}.

Now, list all the whole numbers from 1 to 10 which are not included in the set X.

The whole numbers from 1 to 10 which are not included in set M are 1,3,5,7,8, 9, and 10.

As we know the complement of set X is the set of all the whole numbers from 1 to 10 that are not in set X.

Accordingly, the complement of set X is equal to {1,3,5,7,8,9,10}.
X’ { 1,3,5,7,8,9,10}.
Solved Examples
1. Given Universal Set (U) ={a,b,c,….x,y,z = { a,b,c,d,e} and Y = { E,F,G} , find Y’
Solution: Y’ will include all the letters in english alphabet that are not present in Y. This is represented in the vein diagram below:
Y’ = { a, b, c, d,h, i , j….,x, y, z}
2. If Universal Set (U) = { n n ϵ Z and 6 < n< 7} and B = {Y Y even number; 5 < Y <6}, then what will be the complement of B?
Solution: B’ = { 5,3,1,1,3,5,6}
3. Given U ={ single digit} and B = { 0,1,4,5,6,7,8}, find the complement of B.
Solution: B’ = { 2,3,9}
Hence B’ is the set of all the numbers in universals et (U) that are not included in B. Through setbuilder symbol, we can write: B’ = { xϵ x U and x ∉ B}.
Quiz Time
1. Let the universal set U have all the letters of the English alphabets. What is the complement of the empty set?

U

{a,b,c,d}

ϴ

ϴ – U
2. If E = { 30,31,32,…..45} and D = { multiples of 4} then the complement of aset D is

{ 31.32.35.37}

{ 23,24,25,26,27}

{ 14,42,43,4,,4,5}

{ 30,31,33,34,35,37,38,39,41,42,43,45}
1. What is a Universal Set?
A universal set is a set that includes all the elements or objects of other sets, including its own elements. It is represented by ‘U’. ( There is still no standard symbol of a universal set that can also be represented by any other entities such as ‘V or xi).
For example:
Let us consider three sets named as M, N, and O. The elements of all sets M, N, and O is represented as;
M= { 1,3,6,8}
N = { 2,3,4,5}
O = { 5,8,9}
Find the universal set of all the three sets M, N, and O.
Explanation: As we know, the universal set includes all the elements of a given set. Hence, the universal set of M, N, and O will be,
U = { 1,2,3,4,5,6,8,9}
2. What are the Types of the Complement of a Set?
There are two types of complement of a set.

Relative complement of a set

Absolute complement of a set
Relative complement of a set
The relative complement of a set M in terms of set N is the set of all the elements included in N but not in M. It is also known as the difference of two sets M and N denoted as “MN”
Here, the oval with green color shows that difference is MN
Whereas, the oval with yellow color shows viceversa and i.e. is NM.
Example:
M = { 2,3,4,5,6,7}
N = {3,5,7,9,11,13}
MN = { 2,4,6}
N = { 9,11,14}
MN is not necessarily equivalent to NM.
Relation between set difference
M’N’ = NM
Absolute complement of a set
The absolute complement of a set is defined as the complement of set m i.e. the set of all the elements in Universals et (U) but not in M. It is the relative complement of M in U. So, we can say that complement of M is
M’ = U – M
M’ ={ x : x U, x M}