Chapter 3: Sets, Relations & Functions

Class 11 Applied Mathematics

Master the language of mathematics through logic, collections, and mappings

📚1. Sets and Their Representations

Introduction to Sets

Sets are the fundamental building blocks of modern mathematics. They allow us to group objects together and apply logic to them.

Definitions

Definition

A set is a well-defined collection of distinct objects.

Representation Methods

Roster Form: Listing elements within curly brackets, e.g., $A = {1, 2, 3, 4}$ .
Set-Builder Form: Using a property $P(x)$ to define elements, e.g., $A = {x : x \in N, x < 5}$ .

Types of Sets

⚠️

Empty Set (Null/Void)

A set containing no elements, denoted by

\phi

or {}.

ℹ️

Subsets

Set

A

is a subset of

B

(

A \subseteq B

) if every element of

A

is also an element of

B

Power Set

The collection of all subsets of a set

A

is called the Power Set of

A

, denoted by

P(A)

. If

n(A) = m

, then

n(P(A)) = 2^m

🎨2. Set Operations & Venn Diagrams

Union ($A cup B$)

The union of

A

and

B

(

A \cup B

) is the set of elements which are in

A

or in

B

or in both.

union

Difference ($A - B$)

The difference

A - B

is the set of elements which belong to

A

but not to

B

difference A

Intersection ($A cap B$)

The intersection of

A

and

B

(

A \cap B

) is the set of elements which are common to both

A

and

B

intersection

Complement ($A'$)

The complement of

A

(

A'

A^c

) relative to universal set

\mathbb{U}

is the set of elements in

\mathbb{U}

that are not in

A

Operations on 3 Sets ($A, B, C$)

Using Venn diagrams for three sets allows us to visualize more complex relationships.

Union of 3 Sets

union (3 Sets)

$A cup B cup C$

Intersection of 3 Sets

intersection (3 Sets)

$A cap B cap C$

Laws of Algebra of Sets

Idempotent Laws

A \cup A = A

A \cap A = A

Identity Laws

A \cup \phi = A

A \cap \mathbb{U} = A

Commutative Laws

A \cup B = B \cup A

A \cap B = B \cap A

Associative Laws

(A \cup B) \cup C = A \cup (B \cup C)

Distributive Laws

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

De Morgan's Laws

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

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

Cardinality Formulas

n(A \cup B) = n(A) + n(B) - n(A \cap B)

n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)

🔗3. Relations

Cartesian Product

Before defining relations, we need to understand the product of two sets.

Definition

$A \times B = {(a, b) : a \in A, b \in B}$. If $n(A)=p, n(B)=q$, then $n(A \times B) = pq$.

Relation Definition

A relation is simply a subset of the Cartesian product.

💡

Relation R

A relation

R

from a non-empty set

A

to a non-empty set

B

is a subset of the Cartesian product

A \times B

Domain: The set of all first elements of the ordered pairs in

R

. Range: The set of all second elements.

📈4. Functions

ℹ️

Definition

A relation

f

from

A

B

is a function if every element of

A

has one and only one image in

B

Standard Real Functions

Identity Function

$f(x) = x$

Returns the input as output. Graph is a straight line passing through origin at 45°.

Constant Function

$f(x) = c$

Output remains same regardless of input. Graph is a line parallel to x-axis.

Modulus Function

$f(x) = |x|$

Returns absolute value. $x$ if $x \ge 0$ , $-x$ if $x < 0$ . V-shaped graph.

Signum Function

$f(x) = \frac{|x|}{x}$

Values: 1 if $x>0$ , 0 if $x=0$ , -1 if $x<0$ .

Greatest Integer

$f(x) = [x]$

Returns greatest integer less than or equal to $x$ . Step graph.

Exponential Function

$f(x) = a^x, a > 1$

Curve increasing rapidly for $x>0$ . Passes through (0,1).

Algebra of Functions

Addition

(f + g)(x) = f(x) + g(x)

Subtraction

(f - g)(x) = f(x) - g(x)

Multiplication

(f \cdot g)(x) = f(x) \cdot g(x)

Division

(\frac{f}{g})(x) = \frac{f(x)}{g(x)}, \text{ provided } g(x) \neq 0

Composite Functions ($g circ f$)

💡

Definition

Let

f: A \rightarrow B

and

g: B \rightarrow C

. The composition of

f

and

g

, denoted by

g \circ f

, is defined as

(g \circ f)(x) = g(f(x))

for all

x \in A

1. Generally

g \circ f \neq f \circ g

(Not Commutative). 2.

(h \circ g) \circ f = h \circ (g \circ f)

(Associative).

Composite Function Machine ($g \circ f$)

Input

Process f

Function f

f(x)

Output of f

Process g

Function g

g(f(x))

Final Output

The input x is processed by f first, and then the result f(x) becomes the input for g.

Examples

Example 1

f(x) = 2x

g(x) = x^2

Find $g circ f$:

(g \circ f)(x) = g(f(x)) = g(2x) = (2x)^2 = 4x^2

Note:

(f \circ g)(x) = f(g(x)) = f(x^2) = 2x^2

. Here

g \circ f \neq f \circ g

Example 2

f(x) = \sin x

g(x) = x^2

Find $g circ f$:

(g \circ f)(x) = g(\sin x) = (\sin x)^2 = \sin^2 x

(f \circ g)(x) = f(x^2) = \sin(x^2)

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Chapter 3: Sets, Relations & Functions

Class 11 Applied Mathematics

Master the language of mathematics through logic, collections, and mappings

📚1. Sets and Their Representations

Introduction to Sets

Sets are the fundamental building blocks of modern mathematics. They allow us to group objects together and apply logic to them.

Definitions

Definition

A set is a well-defined collection of distinct objects.

Representation Methods

Roster Form: Listing elements within curly brackets, e.g., $A = {1, 2, 3, 4}$ .
Set-Builder Form: Using a property $P(x)$ to define elements, e.g., $A = {x : x \in N, x < 5}$ .

Types of Sets

⚠️

Empty Set (Null/Void)

A set containing no elements, denoted by

\phi

or {}.

ℹ️

Subsets

Set

A

is a subset of

B

(

A \subseteq B

) if every element of

A

is also an element of

B

Power Set

The collection of all subsets of a set

A

is called the Power Set of

A

, denoted by

P(A)

. If

n(A) = m

, then

n(P(A)) = 2^m

🎨2. Set Operations & Venn Diagrams

Union ($A cup B$)

The union of

A

and

B

(

A \cup B

) is the set of elements which are in

A

or in

B

or in both.

union

Difference ($A - B$)

The difference

A - B

is the set of elements which belong to

A

but not to

B

difference A

Intersection ($A cap B$)

The intersection of

A

and

B

(

A \cap B

) is the set of elements which are common to both

A

and

B

intersection

Complement ($A'$)

The complement of

A

(

A'

A^c

) relative to universal set

\mathbb{U}

is the set of elements in

\mathbb{U}

that are not in

A

Operations on 3 Sets ($A, B, C$)

Using Venn diagrams for three sets allows us to visualize more complex relationships.

Union of 3 Sets

union (3 Sets)

$A cup B cup C$

Intersection of 3 Sets

intersection (3 Sets)

$A cap B cap C$

Laws of Algebra of Sets

Idempotent Laws

A \cup A = A

A \cap A = A

Identity Laws

A \cup \phi = A

A \cap \mathbb{U} = A

Commutative Laws

A \cup B = B \cup A

A \cap B = B \cap A

Associative Laws

(A \cup B) \cup C = A \cup (B \cup C)

Distributive Laws

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

De Morgan's Laws

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

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

Cardinality Formulas

n(A \cup B) = n(A) + n(B) - n(A \cap B)

n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)

🔗3. Relations

Cartesian Product

Before defining relations, we need to understand the product of two sets.

Definition

$A \times B = {(a, b) : a \in A, b \in B}$. If $n(A)=p, n(B)=q$, then $n(A \times B) = pq$.

Relation Definition

A relation is simply a subset of the Cartesian product.

💡

Relation R

A relation

R

from a non-empty set

A

to a non-empty set

B

is a subset of the Cartesian product

A \times B

Domain: The set of all first elements of the ordered pairs in

R

. Range: The set of all second elements.

📈4. Functions

ℹ️

Definition

A relation

f

from

A

B

is a function if every element of

A

has one and only one image in

B

Standard Real Functions

Identity Function

$f(x) = x$

Returns the input as output. Graph is a straight line passing through origin at 45°.

Constant Function

$f(x) = c$

Output remains same regardless of input. Graph is a line parallel to x-axis.

Modulus Function

$f(x) = |x|$

Returns absolute value. $x$ if $x \ge 0$ , $-x$ if $x < 0$ . V-shaped graph.

Signum Function

$f(x) = \frac{|x|}{x}$

Values: 1 if $x>0$ , 0 if $x=0$ , -1 if $x<0$ .

Greatest Integer

$f(x) = [x]$

Returns greatest integer less than or equal to $x$ . Step graph.

Exponential Function

$f(x) = a^x, a > 1$

Curve increasing rapidly for $x>0$ . Passes through (0,1).

Algebra of Functions

Addition

(f + g)(x) = f(x) + g(x)

Subtraction

(f - g)(x) = f(x) - g(x)

Multiplication

(f \cdot g)(x) = f(x) \cdot g(x)

Division

(\frac{f}{g})(x) = \frac{f(x)}{g(x)}, \text{ provided } g(x) \neq 0

Composite Functions ($g circ f$)

💡

Definition

Let

f: A \rightarrow B

and

g: B \rightarrow C

. The composition of

f

and

g

, denoted by

g \circ f

, is defined as

(g \circ f)(x) = g(f(x))

for all

x \in A

1. Generally

g \circ f \neq f \circ g

(Not Commutative). 2.

(h \circ g) \circ f = h \circ (g \circ f)

(Associative).

Composite Function Machine ($g \circ f$)

Input

Process f

Function f

f(x)

Output of f

Process g

Function g

g(f(x))

Final Output

The input x is processed by f first, and then the result f(x) becomes the input for g.

Examples

Example 1

f(x) = 2x

g(x) = x^2

Find $g circ f$:

(g \circ f)(x) = g(f(x)) = g(2x) = (2x)^2 = 4x^2

Note:

(f \circ g)(x) = f(g(x)) = f(x^2) = 2x^2

. Here

g \circ f \neq f \circ g

Example 2

f(x) = \sin x

g(x) = x^2

Find $g circ f$:

(g \circ f)(x) = g(\sin x) = (\sin x)^2 = \sin^2 x

(f \circ g)(x) = f(x^2) = \sin(x^2)

📚1. Sets and Their Representations

Introduction to Sets

Definitions

Definition

Representation Methods

Types of Sets

Empty Set (Null/Void)

Subsets

Power Set

🎨2. Set Operations & Venn Diagrams

Union ($A cup B$)

union

Difference ($A - B$)

difference A

Intersection ($A cap B$)

intersection

Complement ($A'$)

Operations on 3 Sets ($A, B, C$)

Union of 3 Sets

union (3 Sets)

Intersection of 3 Sets

intersection (3 Sets)

Laws of Algebra of Sets

Idempotent Laws

Identity Laws

Commutative Laws

Associative Laws

Distributive Laws

De Morgan's Laws

Cardinality Formulas

🔗3. Relations

Cartesian Product

Definition

Relation Definition

Relation R

📈4. Functions

Definition

Standard Real Functions

Identity Function

Constant Function

Modulus Function

Signum Function

Greatest Integer

Exponential Function

Algebra of Functions

Composite Functions ($g circ f$)

Definition

Composite Function Machine ($g \circ f$)

Examples

❓Practice Problems

Topic-wise Exercises

🔥 CBSE Exam Priority Questions

Most Likely Questions

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📚1. Sets and Their Representations

Introduction to Sets

Definitions

Definition

Representation Methods

Types of Sets

Empty Set (Null/Void)

Subsets

Power Set

🎨2. Set Operations & Venn Diagrams

Union ($A cup B$)

union

Difference ($A - B$)

difference A

Intersection ($A cap B$)

intersection

Complement ($A'$)

Operations on 3 Sets ($A, B, C$)

Union of 3 Sets

union (3 Sets)

Intersection of 3 Sets

intersection (3 Sets)

Laws of Algebra of Sets

Idempotent Laws

Identity Laws

Commutative Laws