Complement of a set, Law of Complement, Property of Complement

Overview of Sets

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U,V...

Now, let us look into the definition that explain complement of set in detail.

Complement of a Set in Set Theory

The complement of a set is one of the most important and frequently used set operations in set theory and discrete mathematics. It helps us identify the elements that do not belong to a particular set when compared to a larger reference set called the universal set.

In simple words, while a set tells us what is included, the set complement tells us what is excluded or missing. This idea is widely used in Venn diagrams, probability, logical reasoning, Boolean algebra, and data analysis problems, where we often need to focus on elements outside a specific group.

Before understanding the definition, laws, and properties of complement of a set, it is important to clearly understand the concepts of sets and the universal set, because the complement is always defined relative to the universal set.

Definition of Complement of a Set

Let $U$ be the universal set and let $B$ be a subset of $U$. The complement of set $B$ is the set of all elements that belong to $U$ but do not belong to $B$.

In other words, it contains everything outside set $B$ but inside the universal set.

Symbol and Notation of Complement of a Set

The complement of a set is commonly represented using the symbols $B'$ or $B^C$, which are read as “B complement”.

Mathematically, it is defined as $B' = \{x \in U \mid x \notin B\}$, meaning all elements in $U$ that are not in $B$.

This can also be written in subtraction form as $B' = U - B$.

Complement of a Set: Examples

Understanding the complement of a set with examples helps in applying the concept easily to solve problems based on universal sets and subsets. Below are a few solved examples that demonstrate how to find the complement of a set.

Example 1:
Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{1, 4, 6, 7, 3, 8\}$
Then, the complement of set $A$ is:
$A' = \{2, 5, 9, 10\}$

Example 2:
Let $U = \{\text{blue, violet, green, yellow, grey, brown, black, white, red}\}$
Let $A = \{\text{violet, green, yellow, white}\}$ and $B = \{\text{grey, brown}\}$
Then:
$A' = \{\text{blue, grey, brown, black, red}\}$
$B' = \{\text{blue, violet, green, yellow, black, white, red}\}$

Example 3:
Let $U = \{x \in \mathbb{N} \mid x \leq 10\}$ and $A = \{x \in \mathbb{N} \mid x \leq 5\}$

Now,
$U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ and $A = \{1, 2, 3, 4, 5\}$
So, $A' = \{6, 7, 8, 9\}$

These examples of complement of a set clearly show how to subtract a subset from its universal set to find the remaining elements.

Complement of Set: Venn Diagram

The Venn diagram for the complement of a set helps visualize all elements in the universal set that do not belong to the given set. In a typical Venn diagram, the universal set is represented by a rectangle, and the set in question is shown as a circle within it.

To illustrate the complement of set $A$, we shade the region outside the circle representing $A$, which includes all elements in $U$ that are not in $A$.

This visual representation makes it easier to understand how $A' = U - A$, and is widely used in logic problems and set theory applications.

Laws of the Complement of a Set

The laws of complement of a set are key rules in set theory that describe the relationship between a set, its complement, the universal set, and the empty set. These properties are essential for simplifying set expressions and solving problems involving set operations and Venn diagrams.

Here are the fundamental complement laws in set theory:

1. Complementation Law:
   The complement of the complement of a set returns the original set.
   $\left(A'\right)' = A$
2. Universal Set Complement Law:
   The complement of the universal set is the empty set.
   $U' = \phi$
3. Empty Set Complement Law:
   The complement of the empty set is the universal set. $\phi' = U$

Properties of Complement of Sets

The properties of complement of sets help in understanding how a set behaves with respect to its complement and the universal set. These properties are foundational in set theory, especially in solving problems using Venn diagrams, union, intersection, and De Morgan’s laws. Below are the main properties explained with examples.

1. Complement Laws

• If $A$ is a subset of the universal set $U$, then $A'$ is also a subset of $U$.
• The union of a set and its complement is the universal set: $A \cup A' = U$
• The intersection of a set and its complement is the empty set: $A \cap A' = \emptyset$

Example:
Let $U = \{1, 2, 3, 4, 5\}$ and $A = \{4, 5\}$.
Then, $A' = \{1, 2, 3\}$
Now, $A \cup A' = \{1, 2, 3, 4, 5\} = U$
And, $A \cap A' = \emptyset$

2. Law of Double Complementation

• The complement of the complement of a set gives the original set: $(A')' = A$
• This means if $A'$ is the complement of $A$, then the complement of $A'$ brings us back to $A$.

Example:
Let $U = \{1, 2, 3, 4, 5\}$ and $A = \{4, 5\}$.
Then, $A' = \{1, 2, 3\}$ and $(A')' = \{4, 5\} = A$

3. Law of Empty Set and Universal Set

• The complement of the universal set is the empty set: $U' = \emptyset$
• The complement of the empty set is the universal set: $\emptyset' = U$

Example:
If $U = \{1, 2, 3, 4, 5\}$, then $U' = \emptyset$ and $\emptyset' = \{1, 2, 3, 4, 5\}$

4. De Morgan’s Laws

• The complement of the union of two sets equals the intersection of their complements: $(A \cup B)' = A' \cap B'$
• The complement of the intersection of two sets equals the union of their complements: $(A \cap B)' = A' \cup B'$

These are known as De Morgan’s Laws and are essential in logic and set theory.

Important Notes on the Complement of a Set

• The complement of a set $A$ is denoted as $A'$ and is defined as: $A' = U - A$
• A set and its complement are always disjoint.
• The complement of the universal set is the empty set: $U' = \emptyset$
• The complement of the empty set is the universal set: $\emptyset' = U$

These properties of set complement are frequently applied in mathematical logic, set operations, and Venn diagram-based questions.

Relationship Between Complement, Union, and Intersection in Set Theory

In set theory and discrete mathematics, the three most important set operations are union, intersection, and complement. These operations are closely related and often work together to help us combine sets, find common elements, or identify elements outside a set.

While union merges elements and intersection finds similarities, the complement of a set focuses on what is missing or excluded relative to the universal set. Understanding the relationship between complement, union, and intersection is essential for solving Venn diagram problems, probability questions, Boolean algebra expressions, and logical reasoning tasks.

Before connecting them, let’s quickly recall what each operation does.

Union of Sets

The union combines all elements from two or more sets.

Mathematically, $A \cup B$ includes elements that belong to $A$ or $B$ or both.

It expands the set.

Intersection of Sets

The intersection keeps only the common elements.

Mathematically, $A \cap B$ includes elements that belong to both $A$ and $B$.

It narrows the set.

Complement of a Set

The complement includes elements that are not in the given set but belong to the universal set.

Mathematically, $A' = U - A$.

It shows what is outside the set.

How Complement Connects with Union and Intersection

Here’s where the real relationship appears. Complement follows special rules when applied to union and intersection. These rules are called De Morgan’s Laws, and they are extremely important in set algebra, logic, and probability.

De Morgan’s Laws (Key Relationship Formulas)

Given below are De Morgan's Laws which are commonly used while solving sets and operations related to sets:

Complement of Union

The complement of a union equals the intersection of complements.

$(A \cup B)' = A' \cap B'$

Meaning: If we exclude everything from both sets together, it’s the same as excluding each set separately and then finding what is common.

Complement of Intersection

The complement of an intersection equals the union of complements.

$(A \cap B)' = A' \cup B'$

Meaning: If we remove the common part, we are left with everything outside either set.

Other Important Properties and Relationships

A few more important relationships are provided as below:

Double Complement Law

Taking complement twice gives the original set.

$(A')' = A$

Complement with Universal Set

$A \cup A' = U$
$A \cap A' = \phi$

So together they cover everything, but share nothing.

Logical Understanding

Think of it like this:

• Union → combine

• Intersection → common

• Complement → outside

Now:

• Complement of union → outside both

• Complement of intersection → outside at least one

It’s basically flipping AND/OR logic.

Comparison in the Sets operations

Solved Examples Based on the Complement of Set

Example 1: Given $\mathrm{n}(\mathrm{U})=10, \mathrm{n}(\mathrm{A})=5, \mathrm{n}(\mathrm{B})=3$ and $n(A \cap B)=2$. A and B are subsets of $U$, then $n(A \cup B)^{\prime}=$

Solution:
Let $U$ be the universal set and $A$ a subset of $U$. Then the complement of $A$ is the set of all elements of $U$ which are not the elements of A. Symbolically, we write A' to denote the complement of $A$ with respect to $U$.
where $A^{\prime}=\{x: x \in U$ and $x \notin A\}$.Obviously $A^{\prime}=U-A$
$ \begin{aligned} & n(A \cup B)=5+3-2=6 \\ & n(A \cup B)^{\prime}=n(U)-n(A \cup B)=10-6=4 \end{aligned} $
Hence, the answer is 4 .

Example 2: Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements, and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. then the percentage of the population who look into advertisements is:

Solution:

Let $P(A)$ and $P(B)$ denote respectively the percentage of the city population that reads newspapers $A$ and $B$.

Let us consider the total percentage to be 100 . Then from the given data, we have

$P(A)=25, \quad P(B)=20, P(A \cap B)=8$

$\therefore$ Percentage of those who read $A$ but not $B$

$P(A \cap \bar{B})=P(A)-P(A \cap B)=25-8=17 \%$
And, Percentage of those who read $B$ but not $A$

$P(\bar{A} \cap B)=P(B)-P(A \cap B)=20-8=12 \%$

If $\mathrm{P}(\mathrm{C})$ denotes the percentage of those who look into an advertisement, then from the given data we obtain

$ \begin{aligned} & \therefore P(C)=30 \% \text { of } P(A \cap \bar{B})+40 \% \text { of } P(\bar{A} \cap B)+50 \% \text { of } P(A \cap B) \\ & \Rightarrow P(C)=\frac{3}{10} \times 17+\frac{2}{5} \times 12+\frac{1}{2} \times 8 \\ & \Rightarrow P(C)=13.9 \% \end{aligned} $
Hence, the answer is 13.9%.

Example 3: If $U=\{1,2,3,4,5\}, A=\{3,4,5\}$ and $B=\{1,2\}$. Then which of the following is true, if $U$ is a universal set of $A$ and $B$?
1) $A \subset B$
2) $A=B$
3) $A=B^{\prime}$
4) None of these

Solution:
Clearly B $=\mathrm{U}-\mathrm{A}$
Hence, $B=A^{\prime}$ and $A=B^{\prime}$
Hence, the answer is the option 3.

Example 4: If A and B are such sets that $A \cup B=U$ is the universal set. Which of the following must be true?
1) $A \cap B=\phi$
2) $A \cup B=A \cap B$
3) $A=B^c$
4) $A \cap U=A$

Solution:

$A \cup A^{\prime}=U$
A and B don't need to be compliment sets. It is only possible that
$A \cap U=A$
Hence, the answer is the option 4.

Example 5: If $A \cup B=U$ and $A \cap B=\phi$, then which of the following is not true?
1) $A^{\prime}=B$

2) $A=B^{\prime}$

3) $A \cap B=B \cap A$

4) $A \cup B=A \cap B$

Solution:
Clearly, $A$ and $B$ are complements of each other.
$\mathrm{A}=\mathrm{B}^{\prime}$ and $\mathrm{A}^{\prime}=\mathrm{B}$, so options (1) and (2) are correct.
Now option (3) is always correct as it is the commutative law.
In option (4), $A \cup B=U$ and $A \cap B=\phi$, so they are not equal.
Hence, the answer is the option 4.

List of Topics Related to the Complement of a Set

Explore key concepts that closely relate to the complement of a set, including foundational topics like set notations, subsets, and set operations. Understanding these topics strengthens your grasp of how sets interact within a universal set.

NCERT Resources

Explore essential NCERT resources for Class 11 Sets, including detailed solutions, clear revision notes, and handpicked exemplar problems. These materials are tailored to build a strong foundational understanding and support preparation for both board exams and competitive entrance tests.

NCERT Solutions for Class 11 Chapter 1 Sets

NCERT Notes for Class 11 Chapter 1 Sets

NCERT Exemplar for Class 11 Chapter 1 Sets

Practice Questions on Complement of a Set

Sharpen your understanding of the complement of a set with focused practice questions designed to reinforce key definitions, properties, and formulas. These practice MCQs are ideal for testing your conceptual clarity and preparing for board exams and entrance tests. Explore the links below to attempt topic-wise MCQs and advance your set theory skills systematically.

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