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Relations and Functions (Class 12)

NCERT Class 12 Mathematics Chapter 1

Relations and Functions for Class 12 Maths and JEE: types of relations, equivalence relations, one-one and onto functions, composition and inverses.

3 topics Updated 8 Nov 2026

For: Class 12 and JEE

In the syllabus: JEE Class 12 syllabus

With your textbook

NCERT Mathematics Part I (Class 12), Chapter 1 “Relations and Functions”, reprint 2026-27.

Open the official NCERT chapter (PDF, 1.4 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Types of relations

    Types of relations for Class 12 Maths and JEE: empty and universal relations, reflexive, symmetric and transitive tests, equivalence relations and classes.

    NCERT Sections 1.1–1.2 21 min read 6 solved examples 15 practice questions

  2. Types of functions

    Types of functions for Class 12 Maths and JEE: one-one, many-one, onto, into and bijective functions, how to prove each, and how to count them.

    NCERT Section 1.3 19 min read 5 solved examples 12 practice questions

  3. Composition of functions and invertible functions

    Composition of functions for Class 12 Maths and JEE: gof and fog, why order matters, invertible functions, finding the inverse, and NCERT examples solved.

    NCERT Section 1.4 16 min read 5 solved examples 10 practice questions

Key formulas in this chapter

Types of relations

  • empty relation: R = φ; universal relation: R = A × A
  • reflexive: (a, a) ∈ R for every a ∈ A
  • symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R
  • transitive: (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R
  • equivalence relation = reflexive + symmetric + transitive
  • equivalence class: [a] = {b ∈ A: b R a}
  • n(A) = n gives N = n2 pairs in A × A: 2N relations, 2N−n of them reflexive

Types of functions

  • one-one: f(x) = f(x) ⇒ x = x
  • onto: range of f = Y; each y ∈ Y equals f(x) for some x ∈ X
  • bijective = one-one and onto
  • finite X: f: X → X is one-one ⇔ f is onto
  • n(A) = m, n(B) = n, m,n > 0: nm functions; n!/(n − m)! one-one (m ≤ n)

Composition and inverse

  • gof(x) = g(f(x)), for x ∈ A
  • gof ≠ fog in general
  • f invertible: some g: Y → X has gof = IX and fog = IY; g = f−1
  • f invertible ⇔ f one-one and onto
  • f−1(f(x)) = x and f(f−1(y)) = y
  • (gof)−1 = f−1og−1

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