Complement of Set 

The complement of a set in mathematics refers to the collection of all elements in the universal set that are not present in the given set. It is an essential concept in set theory, used to identify the "outside" elements of a set relative to the universal set. The complement of set A is denoted by A' or A^c and its formula is A' = U - A. It is essential for solving problems related to the union, intersection, and difference of sets.

1.0Complement of Set Definition

In Set Theory, the complement of a set refers to all the elements that do not belong to the given set but are present in the universal set.

In simple terms, it represents the "outside part" of a set within a universal set.

2.0What is the Complement of a Set?

The complement of a set A (denoted as A' or Aᶜ) contains all elements of the universal set (U) that are not in A.

 $A^{\prime }=x\in U\mid x\notin A$

3.0Complement of Set Formula

Here’s the complement of set formula:

A' = U - A 

Where:

• U = Universal set
• A = Given set
• A' = Complement of A (elements in U but not in A)

4.0Complement of Set with Example

Example:

Let’s say,

• Universal Set: U = {1, 2, 3, 4, 5, 6, 7, 8}
• Set A: A = {2, 4, 6, 8}

Complement of A:

A' = U - A = {1, 3, 5, 7} 

Answer: Complement of A, A', contains {1, 3, 5, 7}.

5.0Complement of Set Difference

In terms of set difference,

A' = U - A 

This shows that the complement of a set is equivalent to the difference between the universal set and the given set.

6.0Complement of Set Properties

Here are some important properties of complement of sets:

1. Double Complement Law: (A')' = A 
2. Law of Universal Set and Empty Set:

$$\begin{gathered} U^{\prime }=\varnothing \\ {\varnothing }^{\prime }=U \end{gathered}$$

3. Complement Laws (De Morgan's Laws):

$$\begin{gathered} (A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime } \\ (A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime } \end{gathered}$$

4. Intersection with Complement:

$$\begin{gathered} A\cap A^{\prime }=\varnothing \\ A\cup A^{\prime }=U \end{gathered}$$

7.0What is the Complement of an A and B Set?

1. Complement of $A\cap B:$

$$\begin{gathered} (A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime } \\ \text{(This is De Morgan’s Law.)} \end{gathered}$$

2. Complement of $A\cup B:$

$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$

8.0What is the Complement of (A ∩ B)?

The complement of the intersection of A and B is: $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$

This means all elements not in both A and B together.

9.0Solved Examples on Complement of Sets 

Example 1: Let the universal set $U=1,2,3,\dots ,10$, A = {2, 4, 6, 8, 10}, and B = {1, 2, 3, 4, 5}. Find _{(A \cup B)}.

Solution:

1. First, find $A\cup B:$

$A\cup B=1,2,3,4,5,6,8,10$

2. Complement of $A\cup BinU:$

$(A\cup B{)}^{\prime }=U-(A\cup B)=7,9$

Answer: $7,9$

Example 2 : $IfU=1,2,3,\dots ,15,A=x\in U\mid x\text{ is a multiple of }3,B=x\in U\mid x\text{ is a multiple of }5,find(A\cap B{)}^{\prime }.$

Solution:

Step 1: Identify sets:

• A = {3, 6, 9, 12, 15} (Multiples of 3)
• B = {5, 10, 15} (Multiples of 5)

Step 2: Intersection: $A\cap B=\{15\}$

Step 3: Complement: 

$(A\cap B{)}^{\prime }=U-15=1,2,3,4,5,6,7,8,9,10,11,12,13,14$

Answer: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

Example 3: Given n(U) = 50, n(A) = 20, n(B) = 30, and $n(A\cap B)=10$, Find $n((A\cup B{)}^{\prime }).$

Solution:

By inclusion-exclusion principle:

$$\begin{gathered} n(A\cup B)=n(A)+n(B)-n(A\cap B) \\ =20+30-10=40 \\ Complement: \\ n((A\cup B{)}^{\prime })=n(U)-n(A\cup B) \\ =50-40=10 \\ Answer:10 \end{gathered}$$

Example 4: If $A^{\prime }\cap B^{\prime }=x\in U\mid x\text{ is an even number less than 10}$ and $U=\{1,2,\dots ,10\}$, find $(A\cup B).$

Solution:

$$\begin{gathered} Given \\ {A}^{\prime }\cap B^{\prime }=\{2,4,6,8,10\} \\ \text{By De Morgan’s Law:} \\ (A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime } \\ \Rightarrow (A\cup B)=U-\{2,4,6,8,10\} \\ A\cup B=\{1,3,5,7,9\} \\ \text{Answer: {1, 3, 5, 7, 9}} \end{gathered}$$

Example 5: If $U=\{x\in \mathbb{B}\mid x\le 20\}$, A = {2, 4, 6, … , 20} (even numbers) and B = {5, 10, 15, 20}, Find $A^{\prime }\cup B^{\prime }.$

Solution:

A' = U - A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

B' = U - B = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19}

$A^{\prime }\cup B^{\prime }\text{includes all elements in either A’ or B’:}$

$A^{\prime }\cup B^{\prime }=\{1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19\}$

Answer:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19}

10.0Practice Questions on Complement of Sets

Question 1: Let the universal set U = {1, 2, 3, …, 12}, A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12}. Find $(A\cap B{)}^{\prime }.$

Question 2: If U = {a, b, c, d, e, f, g}, A = {a, c, e} and B = {b, c, d, e}, $Find(A\cup B{)}^{\prime }.$

Question 3: In a survey, it was found that n(U) = 100, n(A) = 40, n(B) = 30, $n(A\cap B)=10.$
Find $n((A\cap B{)}^{\prime }).$

Question 4: Given U = {1, 2, …, 20}, $A=\{x\in U\mid x\text{ is a prime number}\},$ Find A′.

Question 5: If $A^{\prime }=\{x\in U\mid x\text{ is odd and less than }10\}$ and U = {1, 2, …, 10}, Find the set A.

Answer Key:

$$\begin{gathered} 1.A\cap B{)}^{\prime }=U-(A\cap B)=\{1,3,4,5,7,8,9,10,11\} \\ 2.(A\cup B{)}^{\prime }=U-(A\cup B)=\{f,g\} \\ 3.n((A\cap B{)}^{\prime })=n(U)-n(A\cap B)=100-10=90 \\ 4.A^{\prime }=U-A=1,4,6,8,9,10,12,14,15,16,18,20 \\ 5.A=U-A^{\prime }=2,4,6,8,10 \end{gathered}$$