Relations and Functions – Class 12 Maths Chapter 1 Notes & NCERT PDF

Class 12MathematicsChapter 1NCERT book: Mathematics Part I - Class 12

Notes and a simple summary of Chapter 1 of the NCERT Class 12 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 1: Relations and Functions

Introduction

Chapter 1 of NCERT Class 12 Mathematics (Part I) opens with G. H. Hardy's remark that "there is no permanent place in the world for ugly mathematics". It revisits the relations and functions you met in Class 11 (a relation from A to B is a subset of A × B; a function is a special relation) and now studies types of relations, types of functions, and the composition and inverse of functions. In the rationalised syllabus the chapter has four sections: 1.1 Introduction, 1.2 Types of Relations, 1.3 Types of Functions and 1.4 Composition of Functions and Invertible Function.

The book's opening examples come from everyday life: "is brother of" among students of a boys' school, "is parallel to" among lines, "is congruent to" among triangles, and "the roll number of a student" as a function from students to natural numbers.

Key Concepts

1.2 Types of Relations

For a relation R in a set A:

Empty relation: no element is related to any element, R = φ ⊂ A × A. Example: R = {(a, b) : a − b = 10} on the students of one school, if no two students' ages differ that much.

Universal relation: every element is related to every element, R = A × A.

Reflexive: (a, a) ∈ R for every a ∈ A.

Symmetric: (a₁, a₂) ∈ R ⇒ (a₂, a₁) ∈ R.

Transitive: (a₁, a₂) ∈ R and (a₂, a₃) ∈ R ⇒ (a₁, a₃) ∈ R.

Equivalence relation: reflexive, symmetric and transitive together.

An equivalence relation R on X partitions X into mutually disjoint equivalence classes. The class [a] is the set of all elements related to a. The union of all classes is X, and two classes are either identical or disjoint.

Standard examples from the text:

"Is congruent to" on triangles is an equivalence relation.

"Is perpendicular to" on lines is only symmetric: a line is not perpendicular to itself, and L₁ ⊥ L₂ with L₂ ⊥ L₃ makes L₁ ∥ L₃.

On Z, R = {(a, b) : 2 divides a − b} is an equivalence relation. Its two classes are the even integers and the odd integers.

R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} on {1, 2, 3} is reflexive but neither symmetric nor transitive.

1.3 Types of Functions

One-one (injective): f(x₁) = f(x₂) ⇒ x₁ = x₂. Different elements have different images.

Onto (surjective): every y in the codomain has at least one pre-image, so the range equals the codomain.

Bijective: both one-one and onto.

Book examples to remember:

f : N → N, f(x) = 2x is one-one but not onto (odd numbers have no pre-image).

f : R → R, f(x) = 2x is bijective.

The greatest integer function [x], the modulus function |x| and the signum function on R are neither one-one nor onto.

Characteristic property of finite sets: for a finite set X, a function f : X → X is one-one if and only if it is onto. This fails for infinite sets. For example, f(n) = n + 1 on N is one-one but not onto.

1.4 Composition of Functions and Invertible Function

If f : A → B and g : B → C, then gof : A → C is defined by (gof)(x) = g(f(x)). Apply f first, then g.

In general gof ≠ fog. With f(x) = cos x and g(x) = 3x², gof(x) = 3cos²x but fog(x) = cos 3x².

f : X → Y is invertible if there is g : Y → X with gof = I_X and fog = I_Y. Then g = f⁻¹.

Key theorem: f is invertible if and only if f is one-one and onto. So check bijectivity before finding an inverse.

To find f⁻¹, write y = f(x) and solve for x in terms of y.

Worked Examples

Equivalence check (book Example 5 type). On Z, R = {(a, b) : 2 divides a − b}.

- Reflexive: a − a = 0 is divisible by 2.

- Symmetric: if 2 | (a − b), then 2 | (b − a).

- Transitive: if 2 | (a − b) and 2 | (b − c), then a − c = (a − b) + (b − c) is divisible by 2.

- So R is an equivalence relation, with classes E (even) and O (odd).

Equivalence classes (Example 6). On A = {1, …, 7}, R = {(a, b) : |a − b| is even}. The classes are {1, 3, 5, 7} and {2, 4, 6}. Every element of one class is related to every element of that class, and to none in the other.

Composition (Example 15). Take f : {2, 3, 4, 5} → {3, 4, 5, 9} with f(2) = 3, f(3) = 4, f(4) = f(5) = 5, and g : {3, 4, 5, 9} → {7, 11, 15} with g(3) = g(4) = 7, g(5) = g(9) = 11. Then gof(2) = gof(3) = 7 and gof(4) = gof(5) = 11.

Inverse (Example 17). f : N → Y, f(x) = 4x + 3, where Y is the range of f. Solving y = 4x + 3 gives x = (y − 3)/4, so f⁻¹(y) = (y − 3)/4.

Counting (Examples 22–24).

- The number of one-one functions from {1, 2, 3} to itself is 3! = 6.

- Exactly 1 relation containing (1, 2) and (1, 3) is reflexive and symmetric but not transitive.

- Exactly 2 equivalence relations on {1, 2, 3} contain (1, 2).

Formula / Key-Facts Sheet

Number of relations on a set with n elements: 2^(n²).

Number of one-one (equivalently onto, equivalently bijective) functions from an n-element set to itself: n!.

Invertible ⇔ bijective. For bijections, (gof)⁻¹ = f⁻¹og⁻¹.

For a finite set X, f : X → X is one-one ⇔ onto.

Board Exam Focus

Prove an equivalence relation (3–5 marks). Test reflexive, symmetric and transitive separately, each with a one-line reason, then state the equivalence classes if asked.

One-one/onto proofs. For one-one, assume f(x₁) = f(x₂) and derive x₁ = x₂. For onto, take any y in the codomain, solve for x, and check that x lies in the domain.

MCQs from the Miscellaneous Exercise, such as counting relations or equivalence relations on {1, 2, 3}. Answers: 1 and 2 respectively.

Key Terms

Relation (संबंध)

Reflexive (स्वतुल्य)

Symmetric (सममित)

Transitive (संक्रामक)

Equivalence relation (तुल्यता संबंध)

Equivalence class (तुल्यता वर्ग)

One-one / injective (एकैकी)

Onto / surjective (आच्छादक)

Bijective (एकैकी आच्छादी)

Composition (संयोजन)

Inverse function (प्रतिलोम फलन)

Domain (प्रांत)

Codomain (सहप्रांत)

Range (परिसर)

Common Mistakes

Proving only two of the three properties and calling the relation an equivalence relation.

Confusing codomain with range: onto means range = codomain.

Writing fog when gof is asked. In gof, f acts first.

Finding a formula for f⁻¹ without first proving that f is bijective.

Assuming "one-one ⇔ onto" holds for infinite sets like N.

💡 Key Learning Points

✓Classify relations as empty, universal, reflexive, symmetric, transitive and equivalence relations.
✓Find equivalence classes and understand how an equivalence relation partitions a set.
✓Test functions for one-one, onto and bijective using the book's standard examples.
✓Form compositions gof and fog and recognise that composition is not commutative.
✓Use 'invertible ⇔ bijective' to find inverses such as f(x) = 4x + 3.

👨‍🏫 Teaching Tips

→Model the three properties with arrow diagrams on {1, 2, 3} before moving to infinite sets like Z.
→Use the book's real-life relations (brother of, parallel to, congruent to) to build intuition before formal proofs.
→Run a 'counter-example hunt': students must break symmetry or transitivity with one pair.
→Graph f(x) = 2x, [x], |x| and signum and use horizontal-line reasoning to discuss one-one/onto.
→Practise gof versus fog with f(x) = cos x, g(x) = 3x² side by side.
→Give a proof template for one-one/onto and for equivalence proofs to fix exam presentation.
→Enumerate all relations on a tiny set with students to justify the Miscellaneous Exercise counting answers.

📋 Assessment Questions

Show that R = {(a, b) : 2 divides a − b} on Z is an equivalence relation and name its classes.
Is f : N → N, f(x) = 2x onto? Justify.
If f(x) = cos x and g(x) = 3x², find gof and fog.
Find the inverse of f(x) = 4x + 3 on its range and justify invertibility.
How many equivalence relations on {1, 2, 3} contain (1, 2)?

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Next: Chapter 2 →Inverse Trigonometric Functions