How to Find the Complement of a Set: Step-by-Step Guide
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
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 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?
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}
FAQs on Complement of a Set: Concept, Notation, and Examples
1. What is the complement of a set in Mathematics, as per the NCERT syllabus?
The complement of a set A is the set of all elements that are in the universal set (U) but are not in set A. It is denoted by A' or Ac. For example, if U = {1, 2, 3, 4, 5} and set A = {1, 3}, then the complement of A, written as A', would be {2, 4, 5}.
2. How is the complement of a set denoted and written in set-builder form?
The complement of a set A can be denoted in two common ways: A' (A prime) or Ac (A complement). In set-builder notation, it is formally defined in relation to the universal set U as: A' = {x : x ∈ U and x ∉ A}. This expression reads as 'A complement is the set of all elements x such that x belongs to the universal set U and x does not belong to set A'.
3. How can you represent the complement of a set using a Venn diagram?
In a Venn diagram, the universal set (U) is typically represented by a rectangle. A set A within U is represented by a circle inside this rectangle. The complement of A (A') is represented by the shaded area that is outside the circle of set A but inside the rectangle of the universal set U. This visually shows all the elements that belong to U but not to A.
4. How is the complement of a set different from the difference between two sets (e.g., B - A)?
The key difference lies in the reference set. The complement of a set A (A') is always calculated with respect to the universal set U (i.e., it is U - A). It includes all elements of U that are not in A. In contrast, the difference between two sets (B - A) contains elements that are in set B but not in set A. The complement is a special case of set difference where the first set is always the universal set.
5. What are De Morgan's Laws and why are they important for understanding set complements?
De Morgan's Laws are two fundamental rules that describe the relationship between union, intersection, and complementation. They are crucial for simplifying complex set expressions. The two laws are:
The complement of the union of two sets is the intersection of their complements: (A ∪ B)' = A' ∩ B'.
The complement of the intersection of two sets is the union of their complements: (A ∩ B)' = A' ∪ B'.
These laws provide a method to 'distribute' the complement operator over union or intersection operations, which is essential in both set theory and logic.
6. What happens if you take the complement of an already complemented set, like (A')'?
When you take the complement of a complement, you get the original set back. This is known as the Law of Double Complementation. So, (A')' = A. The first complement (A') includes all elements of U that are not in A. The second complement ((A')') includes all elements of U that are not in A', which by definition are the elements of A itself.
7. What is the complement of the universal set (U) and the empty set (∅)?
This is a concept based on the laws of complementation:
The complement of the universal set (U') is the empty set (∅). This is because there are no elements in U that are not in U.
The complement of the empty set (∅') is the universal set (U). This is because all elements of the universal set are, by definition, not in the empty set.
8. Can you provide a solved example of finding the complement of a set?
Certainly. Let's assume the Universal set for a problem is U = {x : x is a positive integer less than 10}, so U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let set A be the set of even numbers in U, so A = {2, 4, 6, 8}.
To find the complement of A (A'), we must list all the elements that are in U but not in A.
A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
Therefore, A' = {1, 3, 5, 7, 9}.
