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Complement of a Set
The complement of a set is the set that includes all the elements of the universal set that are not present in the given set. Let's say A is a set of all coins which is a subset of a universal set that contains all coins and notes, so the complement of set A is a set of notes (which does not includes coins).
In this article we will discuss in detail the complement of a set, its definition along with properties, solved examples, and practice questions.
1.  What is the Complement of a Set? 
2.  Complement of a Set Venn Diagram 
3.  Properties of Complement of a set 
4.  Complement of a Set Examples 
5.  FAQs on Complement of a Set 
What is the Complement of a Set?
If universal set (U) is having a subset A then the complement of a set A which is represented as A' contains the elements other than the elements of set A. i.e., A' includes the elements of the universal set but not the elements of set A. Mathematically the complement of a set A is written as, A' = {x ∈ U : x ∉ A}. In other words, the complement of a set A is the difference between the universal set and set A. i.e., A' = U  A.
Complement of Set Symbol
The complement of any set is is represented as A', B', C' etc. In other words, we can say, if the universal set is (U) and the subset of the universal set (A) is given then the difference between universal set (U) and the subset of the universal set (A) is the complement of the subset, that is
 A' = U  A [OR]
 A' = {x ∈ U : x ∉ A}
Example of Complement of a Set
The procedure of finding the complement of a set is demonstrated by an example here. If the universal set is all prime numbers up to 25 and set A = {2, 3, 5} then the complement of set A contains elements other than the elements of A.
 Step 1: Check for the universal set and the set for which you need to find the complement. U = {2, 3, 5, 7, 11, 13, 17, 19, 23}, A = {2, 3, 5}.
 Step 2: Subtract, that is (U  A). Here,
U  A = A'
= { ̶2̶,̶ 3̶,̶ 5̶, 7, 11, 13, 17, 19, 23}  { ̶2̶,̶ 3̶,̶ 5̶}
= {7, 11, 13, 17, 19, 23}
Complement of a Set Venn Diagram
For a better understanding look at the complement of a set Venn diagram given below which clearly shows the complement of set A that is A'. Here A' is not part of set A and set A is also not a part of A'. i.e., A and A' are two disjoint sets. Also, A and A' are subsets of U.
 Here, the shaded portion in orange shows the set A
 The shaded portion in white shows the complement of set A (A').
Properties of Complement of a set
Following are the properties of the complement of a set that includes complement laws, the law of double complementation, the law of empty set and universal set, and de Morgan's law.
Complement Laws
 If A is a subset of the universal set then A' is also a subset of the universal set, therefore the union of A and A' is the universal set, represented as A ∪ A’ = U
 The intersection of Set A and A' provides the empty set “∅”, represented as A ∩ A’ = ∅
For example, If U = {1, 2, 3, 4, 5} and A = {4 , 5}, then A' = {1, 2, 3 }. Now, notice that A ∪ A’ = U = {1, 2, 3, 4, 5}. Also, A ∩ A’ = ∅
Law of Double Complementation
 In this law, the complement of the complemented set is the original set, (A')' = A
 The complement of the set A′, where A′ itself is the complement of A, the double complement of A is thus A itself.
In earlier example, U = {1, 2, 3, 4, 5} and A = {4 , 5} then A' = {1 , 2 , 3 }.
The complement of A' = (A')' = {4, 5}, which is equal to set A.
Law of Empty set and Universal Set
 The complement of the universal set is an empty set or null set (∅) and the complement of the empty set is the universal set.
 Since the universal set contains all elements and the empty set contains no elements, therefore, their complement is just opposite to each other, represented as ∅' = U And U' = ∅
In the abovegiven example set U = {1, 2, 3, 4, 5}, we can observe that U' = ∅ (empty set) and ∅' = {1, 2, 3, 4, 5}.
De Morgan’s law
Here are the De Morgan's laws that talk about the complement.
 The complement of the union of two sets is equal to the complement of sets and their intersection. (A U B)’ = A’ ∩ B’ (De Morgan’s Law of Union).
 The complement of the intersection of two sets is equal to the complement of sets and their union. (A ∩ B)’ = A’ U B’ (De Morgan’s Law of Intersection).
Here is an example for proving De Morgan's law, U = {1, 2, 3, 4, 5} and A = {4, 5} and B = {1, 2}. Thus,
De Morgan’s Law of Union: (A U B) = {1, 2, 4, 5} and (A U B)' = {3} and thus, A' ∩ B' = {1, 2, 3} ∩ {3, 4, 5} = {3}. Thus, so (A U B)’ = A’ ∩ B’ = {3}.
De Morgan’s Law of Intersection: (A ∩ B) = ∅ (empty), (A ∩ B)' = {1, 2, 3, 4, 5} and thus, A' U B' = {1, 2, 3 } U {3, 4, 5} = {1, 2, 3, 4, 5}. Thus, (A ∩ B)’ = A' U B'
Important Notes on the Complement of a Set:
 The complement of a set A is denoted by A' and is obtained by subtracting A from the universal set U. i.e., A' = U  A.
 A set and its complement are always disjoint.
 The complement of the universal set is the empty set or null set.
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Complement of a Set Examples

Example 1: If B = { p  p is a multiple of 3, p ∈ N }. Find B' (p ∈ N in the bracket indicates N is the universal set) where N is the set of natural numbers.
Solution:
It is given that
 N = U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, … }.
 B = { p  p is a multiple of 3, p ∈ N } ⇒ B = { 3, 6, 9, 12, 15, … }.
Therefore, the complement of the set B is,
B' = U  B = { 1, 2, 4, 5, 7, 8, 10, 11, … }
Answer: { 1, 2, 4, 5, 7, 8, 10, 11, … }

Example 2: If U is the universal set containing 50 students of class X of a coeducational school and A be the set of all girls and it contains 25 girls. Find the number of elements of the complement of a set of girls.
Solution:
If set A contains all girls then the complement of set A is a set of all boys. The difference between the universal set and the set of all girls is the complement of a set of girls.
Thus, n(A') = n(U)  n(A) = 50  25 = 25. Therefore, the complement of the set contains 25 girls.Answer: 25

Example 3: Find the complement of set A and set B also show (A U B)' = A' ∩ B', where U = {11, 12, 13, 14, 15, 16}, A = {12, 13} and B = {13, 14, 15}?
Solution:
Complement of set A or A' contains elements other than elements of set A .
Therefore, A' = {11, 14, 15, 16}.
Similarly B' = {11, 12, 16}.
Now we will find A' ∩ B' which includes the elements which are in contained A' as well as B'.
So A' ∩ B' = {11, 16} ... (1)
Now, A U B = {12, 13, 14, 15}.
So the complement of A U B or (A U B)' = {11, 16} ... (2)
From (1) and (2), (A U B)' = A' ∩ B'
Answer: Hence proved.
FAQs on Complement of a Set
What is the Complement of a Set?
The complement of set A is defined as a set that contains the elements present in the universal set but not in set A. For example, Set U = {2, 4, 6, 8, 10, 12} and set A = {4, 6, 8}, then the complement of set A, A′ = {2, 10, 12}.
How To Find the Complement of a Set?
If the universal set(U) is given and another set A which contains some elements of the universal set is given, we can find the complement of the set A, represent as A'. The elements that are not part of set A but part of set U will be the elements of A', which is the complement of set A. Here, A' = {x ∈ U: x ∉ A}.
What is the Complement of the Universal Set?
The universal set contains all the possible elements whereas the null set contains no elements at all. Thus, a complement of the universal set is the null set.
What is the Complement of a Set A if Universal Set is the Set of Letters in the English Alphabet and Set A is the Set of Consonants in the English Alphabet?
If the universal set is all alphabets and set A contains all the consonants then the complement of set A, that is A' will be the set of vowels of the English alphabets.
What is A Intersection B Complement?
A intersection B complement is nothing but A complement union B complement. (A ∩ B)’ = A’ U B’. This is known as the De Morgan's law of intersection.
What is the Complement of the Intersection of Sets?
The complement of the intersection of sets is the set of elements that are included in the universal set U but not included in the intersection set. For example, suppose set U = set of natural numbers less than 10 and elements of set X = {1, 2, 5, 6} and set Y = {1, 3, 4, 5, 6}. Thus, the intersection of set X and Y or X ∩ Y = {1, 5, 6} and complement of (X ∩ Y) or (X ∩ Y)' = {2, 3, 4, 7, 8, 9}.
One can easily prove that the complement of the intersection of two sets is equal to the union of the complement of each of the sets.
What is the Complement of an Empty Set or Null Set?
An empty set means there are no elements in the set, so the complement of an empty set or a null set is the universal set containing all the elements.
What is A Union B Complement?
A union B complement is nothing but A complement intersection B complement. (A U B)’ = A’ ∩ B’. This is known as the De Morgan's law of union of sets.
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