Complement of a Set in Set Theory

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}.

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

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

6. 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.

7. 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}