Complement of a Set

What is the Complement of a Set?

In complement of a set if ϴ is considered as the universal set and M is considered as the subset of ϴ. The complement of set M is the set of all the elements of ϴ which are not the elements of M.


We represent the complement of a set M in terms of ϴ as M’


For example, if ϴ = { 1, 2, 3 ,4,5, 6,7}

M = {1, 3, 7}.Find M’

Solution: We can see that 2, 4, 5, 6 are the only elements of  ϴ which do not belong to M.

Hence, M’ = { 2, 4,5, 6}.


Set

A set is a group of well-defined 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

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

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

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

Set X ={ 2,4,6}.

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

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

  3. 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.

  4. 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 set-builder 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?

  1. U

  2. {a,b,c,d}

  3. ϴ

  4. ϴ - U


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

  1. { 31.32.35.37}

  2. { 23,24,25,26,27}

  3. { 14,42,43,4,,4,5}

  4. { 30,31,33,34,35,37,38,39,41,42,43,45}

FAQ (Frequently Asked Questions)

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. 

  1. Relative complement of a set

  2. 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 “M-N”

Here, the oval with green color shows that difference is M-N 

(image will be uploaded soon)

Whereas, the oval with yellow color shows vice-versa and i.e. is N-M.


Example:

M = { 2,3,4,5,6,7}

N = {3,5,7,9,11,13}

M-N = { 2,4,6}

N = { 9,11,14}

M-N is not necessarily equivalent to N-M.

Relation between set difference

M'-N' = N-M


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}