Chapter 1 · Class XII Mathematics · NCERT Exercises

Relations & Functions — Exercises

Reflexive, Symmetric, Transitive & Beyond — Every NCERT Solution Explained

📂 4 Exercise Files 📝 61 Questions 🎯 Board + JEE Level

▶ Start Exercise 1.1 📖 Revise Notes

Exercise Index

①

Exercise 1.1

Types of relations — reflexive, symmetric, transitive, equivalence

16 Q Moderate

②

Exercise 1.2

Types of functions — one-one, onto, bijective

12 Q Moderate

③

Exercise 1.3

Composition of functions; invertible functions

14 Q Hard

★

Miscellaneous

Mixed relations, functions and invertibility proofs

19 Q Hard

4 exercise files · 61 total questions

Chapter Snapshot

JEE MainCBSE BoardsCUET

9Concepts

7Formulas

ModerateDifficulty

4–6%Weightage

Before You Begin

Prerequisites

✓Class XI — Relations & Functions (Ch 2)
✓Set notation and Cartesian products
✓Basic proof-writing habits

Have Ready

🔧NCERT Class XII textbook
🔧Notebook for equivalence-class proofs
🔧Arrow-diagram sketches for functions

Syllabus-wise Topic Map

1.1 IntroductionRecap of relation/function as a special case; why Class XII deepens the study

1.2 Types of RelationsEmpty, universal, reflexive, symmetric, transitive, equivalence relations and equivalence classes

1.3 Types of FunctionsOne-one (injective), onto (surjective), bijective functions; verifying each condition

1.4 Composition of Functions and Invertible Function(g∘f)(x); condition for invertibility; finding f⁻¹

Key Formulae — Recall Before Solving

\(R \text{ equivalence} \iff R \text{ reflexive, symmetric, transitive}\)

\(f:A\to B \text{ one-one} \iff f(x_1)=f(x_2)\Rightarrow x_1=x_2\)

\(f:A\to B \text{ onto} \iff \forall y\in B,\ \exists x\in A: f(x)=y\)

\(f \text{ invertible} \iff f \text{ is bijective}\)

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

NCERT Solving Method

Step 1 — For relations: check reflexive (every a~a), symmetric (a~b⇒b~a), transitive (a~b,b~c⇒a~c) one at a time; never assume all three from one. Step 2 — For functions: prove one-one by assuming f(x1)=f(x2) and deriving x1=x2. Step 3 — Prove onto by taking arbitrary y in codomain and solving for x in domain. Step 4 — Invertibility: confirm bijective first, then construct f⁻¹ algebraically and verify f∘f⁻¹=I.