Class 11 Maths NCERT Solutions Chapter 2 Relations and Functions | Boundless Maths
Chapter 2 Class 11 Maths NCERT Solutions · Unit I · Sets and Functions

Class 11 Maths NCERT Solutions Chapter 2: Relations and Functions

Free, step-by-step NCERT Solutions for all three exercises of this chapter — Cartesian products, relations, functions, standard function graphs and the algebra of real functions — solved the way CBSE awards marks, with the key definitions and formulas right on this page.

4Exercises (incl. Misc.)
36Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 2 — Overview

The Cartesian product A × B is the set of every ordered pair (a, b), with n(A × B) = pq. A relation is any subset of A × B, distinguished by its domain, range and codomain; a function is a special relation where every input has exactly one output. This chapter covers seven standard functions and their graphs — identity, constant, polynomial, rational, modulus, signum and greatest integer — and the algebra of real functions: addition, subtraction, scalar multiplication, multiplication and division.

How the Chapter Builds

One Idea Leads to the Next

1

Cartesian Products of Sets

Ordered pairs and A × B — the raw material every relation and function is built from. Exercise 2.1.

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2

Relations

A relation is a subset of A × B — domain, codomain and range, shown with arrow diagrams. Exercise 2.2.

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3

Functions

A relation where every input has exactly one output. Real valued and real functions. Exercise 2.3.

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4

Standard Functions & Graphs

Identity, constant, polynomial, rational, modulus, signum and greatest integer functions. §2.4.1.

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5

Algebra of Real Functions

f + g, f − g, αf, fg and f/g — combining two functions on a common domain. Miscellaneous Exercise.

Quick Reference

Important Formulas — Chapter 2

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.

Cartesian Products of Sets (§2.2)

Cartesian product

P\times Q = \{(p,q) : p\in P,\, q\in Q\}

Every element of P paired with every element of Q, in that order. P × Q = φ if either P or Q is empty.

Equality of ordered pairs

(a,b)=(x,y) \iff a=x \text{ and } b=y

Both corresponding elements must match — order matters, so (a, b) ≠ (b, a) unless a = b.

Size of A × B

n(A)=p,\; n(B)=q \;\Rightarrow\; n(A\times B)=pq

In general A × B ≠ B × A, though both have the same number of elements.

Relations (§2.3)

Relation

R \subseteq A\times B

Any subset of A × B, usually described by a rule linking the first and second elements of each pair.

Domain, range, codomain

\text{Domain}=\{x : (x,y)\in R\} \quad \text{Range}=\{y : (x,y)\in R\}

The codomain is the whole of B that R is defined into — range ⊆ codomain always, but need not be equal.

Number of relations

n(A)=p,\;n(B)=q \;\Rightarrow\; 2^{pq} \text{ relations from } A \text{ to } B

Every subset of A × B is a valid relation, and a set with pq elements has 2^(pq) subsets.

Functions & Standard Graphs (§2.4)

Function

f:A\to B,\;\; \forall\, a\in A\;\; \exists! \, b\in B \text{ with } f(a)=b

Every element of the domain A has exactly one image — no repeats of the first element with a different second element.

Modulus function

f(x)=|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}

Domain R, range [0, ∞) — the graph is a V-shape with vertex at the origin.

Signum function

f(x)=\begin{cases}1,&x>0\\0,&x=0\\-1,&x<0\end{cases}

Domain R, range {−1, 0, 1} — indicates only the sign of x, never its magnitude.

Greatest integer function

f(x)=[x] = \text{greatest integer} \le x
[x] = −1 for −1 ≤ x < 0, [x] = 0 for 0 ≤ x < 1, and so on — the graph is a staircase.

Algebra of Real Functions (§2.4.2)

Sum, difference, scalar

(f\pm g)(x)=f(x)\pm g(x) \qquad (\alpha f)(x)=\alpha\, f(x)

Defined pointwise on the common domain of f and g; α is any real scalar.

Product & quotient

(fg)(x)=f(x)\,g(x) \qquad \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)},\; g(x)\ne 0

The quotient is defined only where g(x) ≠ 0 — always state this restriction explicitly.

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 askingUse thisWhy
List every ordered pair from two given setsCartesian productP × Q pairs every element of P with every element of Q, in that order (§2.2).
Find unknowns from two ordered pairs said to be equalEquality of ordered pairsMatch first elements together and second elements together, then solve (§2.2).
Describe a subset of A × B by a rule between x and yRelation, roster or set-builderState the rule, then list domain, range and codomain separately (§2.3).
Decide if a given set of ordered pairs is a functionFunction testCheck no first element repeats with a different second element (§2.4).
Find where a formula for f(x) is actually definedDomain of a functionExclude values that make a denominator 0 or a square root negative (§2.4).
Identify which named function a graph or rule matchesStandard functionsMatch the shape/rule to identity, modulus, signum, greatest integer, etc. (§2.4.1).
Combine two functions given as f(x) and g(x)Algebra of functionsApply pointwise on the common domain; watch for g(x) ≠ 0 in a quotient (§2.4.2).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all three exercises.

  • Assuming A × B = B × A — the two sets have the same number of elements, but the ordered pairs themselves are generally different, since order matters in a Cartesian product.
  • Confusing a relation with a function — every function is a relation, but a relation is a function only if every element of the domain has exactly one image. A single first element repeating with two different second elements disqualifies it.
  • Mixing up range and codomain — the codomain is the entire target set B, fixed by the definition of the function; the range is only the elements of B that actually get hit. They coincide only for onto functions.
  • Forgetting domain restrictions when a function is a formula — a denominator can never be zero, and an expression under a square root can never be negative. Always state the excluded values explicitly rather than assuming the domain is all of R.
  • Applying the modulus, signum, or greatest integer definitions carelessly at the boundary point — for [x], remember the interval is closed on the left and open on the right (n ≤ x < n + 1), so [x] at an integer equals that integer itself, not one less.
  • Adding, multiplying or dividing two functions without checking their domains agree — f + g, fg and f/g are only defined on the intersection of the domains of f and g, and f/g additionally excludes points where g(x) = 0.
  • Treating the number of relations and the number of functions from A to B as the same count — there are 2^(pq) relations from A to B, but far fewer of those subsets actually qualify as functions.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 2 — Choose an Exercise

2.1

Exercise 2.1

Cartesian products of sets — ordered pairs, equality of ordered pairs, and A × B · 10 questions

Solve Exercise 2.1 →
2.2

Exercise 2.2

Relations — domain, codomain and range, roster and set-builder form, arrow diagrams · 9 questions

Solve Exercise 2.2 →
2.3

Exercise 2.3

Functions — function tests, domain and range of real functions · 5 questions

Solve Exercise 2.3 →
M

Miscellaneous Exercise

Function vs. relation proofs, domain-finding, and the algebra of functions, tying the chapter together · 12 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Relations and Functions — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →
Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 2, Relations and Functions.

How many exercises are there in Chapter 2, Relations and Functions?
There are three main exercises — 2.1 (Cartesian Products of Sets, 10 questions), 2.2 (Relations, 9 questions) and 2.3 (Functions, 5 questions) — plus a Miscellaneous Exercise of 12 questions covering the algebra of functions and proofs about relations, totalling 36 questions.
What is the difference between a relation and a function?
A relation from set A to set B is any subset of the Cartesian product A × B, with no restrictions at all — one input can be paired with several outputs. A function is a special kind of relation where every element of A has exactly one image in B; no first element is repeated with a different second element. Every function is therefore a relation, but not every relation is a function.
What is the difference between the range and the codomain of a function?
The codomain is the entire set B that a function f: A → B is defined into — it is fixed in advance, whether or not every element of B actually gets used. The range is the set of values the function actually outputs, i.e. the set of images of every element of A. The range is always a subset of the codomain, and the two are equal only when the function is onto.
How many relations can be defined from a set A to a set B?
A relation from A to B is any subset of A × B. If n(A) = p and n(B) = q, then n(A × B) = pq, and since a set with k elements has 2k subsets, the total number of possible relations from A to B is 2pq.
Where can I find the official NCERT textbook for this chapter?
Relations and Functions is Chapter 2 of the NCERT Class 11 Mathematics textbook, 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.
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