Class 12 Maths NCERT Solutions Chapter 1 Relations and Functions | Boundless Maths

Chapter 1 Class 12 Maths NCERT Solutions · Unit I · Relations and Functions

Class 12 Maths NCERT Solutions Chapter 1: Relations and Functions

Free, step-by-step NCERT Solutions for both exercises of this chapter — types of relations (reflexive, symmetric, transitive and equivalence relations), equivalence classes, and types of functions (one-one, onto and bijective) — plus the Miscellaneous Exercise covering composition of functions and invertibility. Solved the way CBSE awards marks, with the key definitions and the mistakes that cost students marks every year, right on this page.

Relations and Functions is the opening chapter of Class 12 Maths, and it's really two chapters in one: the first half is about relations — precise, testable properties like reflexivity and symmetry that build up to the single most important idea in the chapter, equivalence relations. The second half moves to functions, sharpening the informal idea from Class 11 into the formal tests for one-one and onto that everything from composition to invertibility depends on.

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Class 12 Maths NCERT Solutions Chapter 1 — Overview

This Class 12 Maths NCERT Solutions Chapter 1 hub covers Relations and Functions, which formalises ideas you first met informally in Class 11. A relation from set A to set B is defined precisely as any subset of A × B — no pattern or rule required, just a collection of ordered pairs. From there, the chapter studies three key properties a relation can have on a single set: reflexive (every element relates to itself), symmetric (the relation works both ways), and transitive (related pairs chain together). A relation with all three properties is an equivalence relation, and every equivalence relation splits its set into disjoint equivalence classes — a genuinely powerful idea that reappears throughout higher mathematics.

The second half of the chapter turns to functions. You'll pin down exactly what makes a function one-one (injective) — distinct inputs never share an output — and onto (surjective) — every element of the codomain gets hit. A function that's both is bijective, and bijective functions are exactly the ones that have a genuine inverse. The Miscellaneous Exercise ties this together with composition of functions (gof) and shows how to prove a function is invertible without necessarily finding the inverse itself, just by checking it's one-one and onto.

How the Chapter Builds

One Idea Leads to the Next

Types of Relations

Reflexive, symmetric and transitive properties on a set — the definitions everything else builds on. Exercise 1.1.

→

Equivalence Relations & Classes

A relation with all three properties partitions its set into disjoint equivalence classes. Exercise 1.1.

→

Types of Functions

One-one (injective) and onto (surjective); a function that's both is bijective. Exercise 1.2.

→

Composition & Invertibility

gof(x) = g(f(x)); a function is invertible if and only if it's bijective. Miscellaneous Exercise.

Quick Reference

Important Formulas — Chapter 1

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Relations and Functions.

Types of Relations (§1.2)

Reflexive, symmetric, transitive

\text{Reflexive: } (a,a)\in R \; \forall\, a\in A

\text{Symmetric: } (a_1,a_2)\in R \Rightarrow (a_2,a_1)\in R

\text{Transitive: } (a_1,a_2),(a_2,a_3)\in R \Rightarrow (a_1,a_3)\in R

All three must hold for every element / every applicable pair — one exception anywhere disproves the property.

Equivalence relation

R \text{ is reflexive} + \text{symmetric} + \text{transitive}

An equivalence relation splits its set into disjoint equivalence classes whose union is the whole set.

Empty & universal relations

R=\phi \subset A\times A \quad\text{(empty)} \qquad R = A\times A \quad\text{(universal)}

The universal relation is always an equivalence relation; the empty relation is one only when A itself is empty.

Types of Functions (§1.3)

One-one (injective)

f(x_1)=f(x_2) \Rightarrow x_1=x_2 \;\; \forall\, x_1,x_2\in X

Distinct inputs must always give distinct outputs. Otherwise f is called many-one.

Onto (surjective)

\forall\, y\in Y,\; \exists\, x\in X \text{ such that } f(x)=y

Every element of the codomain must be hit by some element of the domain. Equivalently, Range(f) = Y.

Bijective

f \text{ is one-one} \;\text{and}\; f \text{ is onto}

For a finite set X, a one-one map f : X → X is automatically onto, and vice versa — this shortcut fails for infinite sets.

Composition & Invertibility (Miscellaneous)

Composition of functions

(g\circ f)(x) = g(f(x)),\;\; \forall\, x\in A

Apply f first, then g. In general, gof ≠ fog — composition is not commutative.

Invertible function

f \text{ invertible} \iff \exists\, g \text{ with } g\circ f = I_X \text{ and } f\circ g = I_Y

f is invertible if and only if f is bijective — proving one-one and onto is enough, without constructing f⁻¹.

Decision Guide

Which Relations / Functions Concept Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is asking	Use this	Why
Check if a single element relates to itself, for every element	Reflexive test	(a,a) must be in R for every a — check ALL elements, not a sample (§1.2).
Check if the relation "works both ways"	Symmetric test	(a,b) ∈ R must force (b,a) ∈ R (§1.2).
Check if related pairs "chain together"	Transitive test	(a,b),(b,c) ∈ R must force (a,c) ∈ R (§1.2).
Prove a relation partitions a set into disjoint groups	Equivalence relation	Prove reflexive, symmetric and transitive together (§1.2).
Show two different inputs never give the same output	One-one (injective) test	f(x₁)=f(x₂) ⟹ x₁=x₂ (§1.3).
Show every element of the codomain is "hit"	Onto (surjective) test	For every y, find some x with f(x)=y (§1.3).
Show a function has a genuine inverse	Prove bijective	Show one-one AND onto — you don't need to find f⁻¹ itself (Miscellaneous).

Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across both exercises.

Checking reflexivity, symmetry or transitivity on only a few elements instead of every element of the set — a single counterexample anywhere disproves the property, but proving it requires checking universally.
Assuming a relation is reflexive just because it "looks" well-behaved — the three properties are independent; a relation can be symmetric and transitive but still fail to be reflexive (the classic trap: the empty relation on a non-empty set).
Confusing one-one with onto — a function can be one-one without being onto, and vice versa. Always check both separately; never assume one implies the other.
Applying the "finite set" shortcut (one-one ⟺ onto) to an infinite-set function — this equivalence only holds when domain and codomain are the same finite set. f: N → N, f(x) = 2x is one-one but not onto, precisely because N is infinite.
Mixing up gof and fog — composition is not commutative in general. Always compose in the order given, applying the innermost function first.
Trying to actually find f⁻¹ when the question only asks you to prove f is invertible — proving f is bijective is enough; you don't need to construct the inverse unless it's explicitly asked for.
Forgetting that equivalence classes must be pairwise disjoint and their union must be the entire set — a common trap in "find the equivalence class" questions.

Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 1 — Choose an Exercise

1.1

Exercise 1.1

Types of relations — reflexive, symmetric, transitive and equivalence relations, plus equivalence classes · 16 questions

Solve Exercise 1.1 →

1.2

Exercise 1.2

Types of functions — one-one, onto and bijective functions · 12 questions

Solve Exercise 1.2 →

Miscellaneous Exercise

Composition of functions, invertibility, and equal functions · 7 questions

Solve Miscellaneous →

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Every formula for Relations and Functions — plus every other Class 12 Maths chapter — in one printable PDF.

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Common Questions

Frequently Asked Questions

Quick answers from Class 12 Maths NCERT Solutions Chapter 1, Relations and Functions.

How many exercises are there in Chapter 1, Relations and Functions?

There are two main exercises — 1.1 (Types of Relations, 16 questions) and 1.2 (Types of Functions, 12 questions) — plus a Miscellaneous Exercise of 7 questions covering composition of functions and invertibility, totalling 35 questions. This is the shortest chapter in the syllabus, since the binary operations section has been removed from the current CBSE syllabus.

What is the difference between a relation and a function?

Every function is a relation, but not every relation is a function. A relation is any subset of A × B, with no restrictions at all. A function additionally requires that every element of the domain maps to exactly one element of the codomain — a relation can pair one input with several outputs, but a function cannot.

What are the three properties of an equivalence relation?

A relation must be reflexive (every element relates to itself), symmetric (if a relates to b, then b relates to a) and transitive (if a relates to b and b relates to c, then a relates to c), all three at once, to be an equivalence relation.

What is the difference between a one-one and an onto function?

One-one (injective) means distinct inputs always give distinct outputs. Onto (surjective) means every element of the codomain is the output of some input. A function needs both properties to be bijective, and only bijective functions have a genuine inverse.

Where can I find the official NCERT textbook for this chapter?

Relations and Functions is Chapter 1 of the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.