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Short Intro

In this post, students can find complete step-by-step solutions for the Miscellaneous Exercise of Class 12 Maths Chapter 1 – Relations and Functions. This exercise includes advanced concepts of equivalence relations, one-one and onto functions, injective mappings, and counting functions based on the latest NCERT syllabus.

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Quick Information Box

Item	Details
Board	NCERT / CBSE
Class	12
Subject	Mathematics
Chapter	Relations and Functions
Exercise	Miscellaneous Exercise
Main Topics	Relations & Functions

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Concepts Used (Topics Covered)

• One-One Functions
• Onto Functions
• Bijective Functions
• Equivalence Relations
• Reflexive Relations
• Symmetric Relations
• Transitive Relations
• Power Set Relations
• Counting Functions
• Equality of Functions

The chapter discusses different types of relations and functions along with injective and surjective mappings.

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Important Formulas

One-One Function

f(x₁) = f(x₂) ⇒ x₁ = x₂

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Onto Function

For every y ∈ Y, there exists x ∈ X such that f(x) = y

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Equivalence Relation

A relation which is reflexive, symmetric and transitive.

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Function Given in Question 1

$f(x)=\frac{x}{1+|x|}$

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Questions & Step-by-step Solutions

Question 1

Show that the function

$f(x)=\frac{x}{1+|x|}$

is one-one and onto.

Solution

Suppose:

f(x₁) = f(x₂)

Then:

x₁/(1+|x₁|) = x₂/(1+|x₂|)

After simplification:

x₁ = x₂

Hence function is one-one.

Now let:

y ∈ (−1,1)

Choose:

x = y/(1−|y|)

Then:

f(x)=y

Hence function is onto.

Therefore:

f is bijective.

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Question 2

Show that

$f(x)=x^3$

is injective.

Solution

Suppose:

f(x₁)=f(x₂)

Then:

x₁^3=x₂^3

Taking cube roots:

x₁=x₂

Therefore:

f is injective.

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Question 3

Relation on Power Set

Given:

ARB iff A ⊂ B

Solution

For equivalence relation, relation must be reflexive, symmetric and transitive.

Here:

A ⊂ A

is true.

Hence reflexive.

But:

A ⊂ B

does not imply:

B ⊂ A

Therefore not symmetric.

Hence:

R is not an equivalence relation.

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Question 4

Number of onto functions from {1,2,3,…,n} to itself

Solution

For finite sets:

Every onto function from a set to itself is one-one.

Hence total onto functions are equal to total permutations.

Therefore:

Number of onto functions = n!

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Question 5

Check whether functions f and g are equal.

Given:

$f(x)=x^2-x$

and

$g(x)=2\left|\frac{x}{2}\right|-1$

Solution

Evaluate functions for all elements of:

A={−1,0,1,2}

For f(x)

x	f(x)
−1	2
0	0
1	0
2	2

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For g(x)

x	g(x)
−1	−2
0	−1
1	0
2	1

Since outputs differ:

f ≠ g

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Question 6

Number of relations containing (1,2) and (1,3) which are reflexive and symmetric but not transitive

Solution

Required answer:

(B) 2

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Question 7

Number of equivalence relations containing (1,2)

Solution

Possible equivalence relations are:

1. Relation where 1 and 2 belong to same class
2. Universal relation

Hence:

(B) 2

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Common Mistakes

• Confusing subset with proper subset
• Forgetting onto condition
• Not checking symmetry separately
• Mistakes in proving injective functions
• Ignoring co-domain while checking onto

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Exam Tips

• Always verify reflexive, symmetric and transitive properties separately.
• Use counterexamples to disprove properties.
• Remember: bijective = one-one + onto.
• Practice function mappings carefully.

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Practice MCQs

MCQ 1

A bijective function is:

A. One-one only
B. Onto only
C. Both one-one and onto
D. None

Answer:

C. Both one-one and onto

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MCQ 2

The number of onto functions from a finite set to itself equals:

A. n
B. n²
C. n!
D. 2ⁿ

Answer:

C. n!

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MCQ 3

Relation:

A ⊂ B

is:

A. Symmetric
B. Reflexive only
C. Equivalence relation
D. Universal relation

Answer:

B. Reflexive only

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FAQ Section

What is an injective function?

A function where different inputs produce different outputs.

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What is a bijective function?

A function which is both one-one and onto.

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What is an equivalence relation?

A relation which is reflexive, symmetric and transitive.

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What is the number of onto functions from a finite set to itself?

n!

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