CBSE Class 12 Relations and Functions - Complete Guide with Types, Composition & Binary Operations

In short: Relations and Functions (Chapter 1) carries 5 marks in the CBSE Class 12 Maths board exam. It covers types of relations (reflexive, symmetric, transitive, equivalence), types of functions (one-one/injective, onto/surjective, bijective), composition of functions, inverse of a function, and binary operations. As the very first chapter of Class 12 Maths, it sets the tone for the entire year — and it is more definition-heavy and proof-based than most students expect. This guide gives you every concept, every proof technique, and the exact strategies to ace this foundational chapter.

Introduction — The First Chapter Sets the Foundation

Relations and Functions belongs to Unit I (Relations and Functions) of the CBSE Class 12 Maths syllabus. The entire unit carries 7 marks out of 80 in the theory paper (Chapter 1 contributes ~5 marks, and Chapter 2 — Inverse Trigonometric Functions — contributes ~2 marks). This chapter is unique because it is less about computation and more about understanding definitions and constructing logical arguments. Questions ask you to prove that a relation is an equivalence relation or show that a function is bijective. Students who are comfortable with Calculus and Algebra often find this chapter challenging because it requires a different kind of thinking — systematic verification of properties. For the complete marks breakdown, check the CBSE Class 12 Maths chapter-wise marks weightage.

Core Concepts, Formulas & Memory Tricks

Relations — Quick Review

A relation R from set A to set B is a subset of A × B. If A = B, we say R is a relation on set A.

Domain: Set of all first elements in R Range: Set of all second elements in R Codomain: Set B (which may be larger than the range)

Types of Relations

1. Empty Relation (∅): No element is related to any element. R = ∅.

2. Universal Relation: Every element is related to every element. R = A × A.

3. Reflexive Relation: Every element is related to itself. Condition: (a, a) ∈ R for all a ∈ A

Example: R = {(1,1), (2,2), (3,3), (1,2)} on A = {1,2,3} is reflexive because (1,1), (2,2), (3,3) are all present.

How to check: Ensure that every element of the set appears as (a,a) in the relation.

4. Symmetric Relation: If a is related to b, then b is related to a. Condition: If (a, b) ∈ R, then (b, a) ∈ R

Example: R = {(1,2), (2,1), (1,1)} on A = {1,2,3} is symmetric because every (a,b) has a corresponding (b,a).

How to check: For every pair (a,b) in R where a≠b, verify that (b,a) is also in R.

5. Transitive Relation: If a is related to b and b is related to c, then a is related to c. Condition: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

Example: R = {(1,2), (2,3), (1,3)} on A = {1,2,3} is transitive because (1,2) and (2,3) are in R, and (1,3) is also in R.

How to check: Look at all chains (a,b) and (b,c) — verify that (a,c) exists in each case.

6. Equivalence Relation: A relation that is reflexive, symmetric, AND transitive.

Memory trick — "RST" for equivalence: An equivalence relation must satisfy all three: Reflexive, Symmetric, Transitive. Remember "RST" — like "reset."

Equivalence Classes

If R is an equivalence relation on set A and a ∈ A, the equivalence class of a is:

[a] = {x ∈ A : (x, a) ∈ R}

All elements related to a form its equivalence class. Equivalence classes partition the set A into non-overlapping groups.

Example: On Z (integers), the relation "a R b if a-b is divisible by 3" is an equivalence relation. The equivalence classes are:

• [0] = {..., -6, -3, 0, 3, 6, ...}
• [1] = {..., -5, -2, 1, 4, 7, ...}
• [2] = {..., -4, -1, 2, 5, 8, ...}

Types of Functions

1. One-one (Injective) Function: Different elements have different images. If f(a₁) = f(a₂), then a₁ = a₂.

Equivalently: If a₁ ≠ a₂, then f(a₁) ≠ f(a₂).

How to prove a function is one-one:

• Assume f(a₁) = f(a₂) and show that a₁ = a₂
• OR show that f is strictly increasing or strictly decreasing (using derivatives if applicable)

2. Onto (Surjective) Function: Every element in the codomain has a pre-image. For every b ∈ B, there exists a ∈ A such that f(a) = b.

How to prove a function is onto:

• Take an arbitrary element y in the codomain
• Solve f(x) = y for x
• Show that x belongs to the domain

3. Bijective Function (One-one AND Onto): Every element in the codomain is mapped to by exactly one element of the domain.

Important: Only bijective functions have inverses.

Memory trick — "One-one = no sharing, Onto = no one left out, Bijective = perfect matching":

• One-one: no two domain elements share the same image
• Onto: no codomain element is left without a pre-image
• Bijective: perfect one-to-one correspondence between domain and codomain

Composition of Functions

If f: A → B and g: B → C, then the composition g ∘ f: A → C is defined as:

(g ∘ f)(x) = g(f(x))

Key properties:

• Composition is NOT commutative: g ∘ f ≠ f ∘ g in general
• Composition is associative: (h ∘ g) ∘ f = h ∘ (g ∘ f)
• If f and g are one-one, then g ∘ f is one-one
• If f and g are onto, then g ∘ f is onto
• If f and g are bijective, then g ∘ f is bijective

How to compute: Work from inside out. First apply f, then apply g to the result.

Example: f(x) = 2x+3, g(x) = x². (g ∘ f)(x) = g(f(x)) = g(2x+3) = (2x+3)² (f ∘ g)(x) = f(g(x)) = f(x²) = 2x²+3

Note: g ∘ f ≠ f ∘ g.

Inverse of a Function

If f: A → B is a bijection, then f⁻¹: B → A exists and satisfies:

f⁻¹(y) = x if and only if f(x) = y

Properties:

• f⁻¹ ∘ f = I_A (identity on A)
• f ∘ f⁻¹ = I_B (identity on B)
• f⁻¹ exists if and only if f is bijective

How to find the inverse:

1. Let y = f(x)
2. Solve for x in terms of y
3. Replace y with x in the result to write f⁻¹(x)

Example: f(x) = (4x+3)/(6x-4) Let y = (4x+3)/(6x-4) y(6x-4) = 4x+3 → 6xy - 4y = 4x+3 → 6xy - 4x = 4y+3 x(6y-4) = 4y+3 → x = (4y+3)/(6y-4) f⁻¹(x) = (4x+3)/(6x-4)

Notice: f⁻¹ = f in this case (the function is its own inverse).

Binary Operations

A binary operation * on set A is a function *: A × A → A.

Properties to check:

• Closure: a * b ∈ A for all a, b ∈ A
• Commutative: a * b = b * a for all a, b
• Associative: (a * b) * c = a * (b * c) for all a, b, c
• Identity element e: a * e = e * a = a for all a
• Inverse of a: a * a⁻¹ = a⁻¹ * a = e

Common binary operations in board exams:

• a * b = a + b - ab on a given set
• a * b = (a + b)/2 on rationals
• a * b = LCM(a, b) or GCD(a, b) on natural numbers

Memory trick: A binary operation takes two elements and returns one element in the same set. Think of it as a "rule" for combining two things — like addition combines two numbers into one number.

Board Exam Question Types

Type 1: Prove that a Relation is an Equivalence Relation (5 marks)

This is the most common 5-mark question from this chapter. You will be given a relation (usually on Z or N) and asked to prove it is an equivalence relation.

Approach:

1. State the three conditions: Reflexive, Symmetric, Transitive
2. Prove Reflexive: Show (a,a) ∈ R for all a. Take any a in the set and verify the defining condition holds when both elements are a.
3. Prove Symmetric: Assume (a,b) ∈ R and show (b,a) ∈ R.
4. Prove Transitive: Assume (a,b) ∈ R and (b,c) ∈ R, and show (a,c) ∈ R.

Example: R on Z defined by a R b iff (a-b) is divisible by 3.

• Reflexive: a-a = 0 = 3×0, divisible by 3 ✓
• Symmetric: If a-b = 3k, then b-a = -3k = 3(-k), divisible by 3 ✓
• Transitive: If a-b = 3k and b-c = 3m, then a-c = (a-b)+(b-c) = 3k+3m = 3(k+m), divisible by 3 ✓

Type 2: Show that a Function is One-one and/or Onto (3-5 marks)

Approach for one-one: Assume f(x₁) = f(x₂), simplify, show x₁ = x₂. Approach for onto: Take arbitrary y in codomain, find x such that f(x) = y, verify x is in domain.

Example: Show that f: R→R defined by f(x) = 3x+7 is bijective.

• One-one: f(x₁) = f(x₂) → 3x₁+7 = 3x₂+7 → x₁ = x₂ ✓
• Onto: For any y ∈ R, let x = (y-7)/3. Then f(x) = 3(y-7)/3+7 = y ✓
• Since f is both one-one and onto, it is bijective ✓

Type 3: Find the Inverse of a Function (3 marks)

Approach: Verify the function is bijective first (or state that it is given to be bijective). Then solve y = f(x) for x and express x as a function of y.

Type 4: Composition of Functions (2-3 marks)

Approach: Compute (g ∘ f)(x) = g(f(x)) by substituting f(x) into g. Similarly for (f ∘ g)(x). Show that g ∘ f ≠ f ∘ g if required.

Type 5: Binary Operations (2-3 marks)

Common questions:

• Check if a given operation is commutative/associative
• Find the identity element
• Find the inverse of an element under the given operation

Approach: Use the definitions directly. For commutativity, check if ab = ba. For identity, find e such that ae = a. For inverse, find b such that ab = e.

Common Mistakes That Cost Marks

1. Checking only some elements for reflexive/symmetric/transitive. When proving a property, you must prove it for ALL elements, not just a few examples. Use general variables (a, b, c) and algebraic reasoning, not specific numbers. Checking (1,1) and (2,2) alone does not prove reflexivity — you need to show (a,a) ∈ R for every a in the set.

2. Confusing one-one with onto. One-one means different inputs give different outputs. Onto means every output is achieved. These are independent properties — a function can be one-one but not onto, onto but not one-one, both, or neither. Read the question carefully to know which property is being asked.

3. Forgetting to verify that x is in the domain when proving onto. Finding x such that f(x) = y is not enough — you must also show that this x belongs to the domain. If the domain is R⁺ (positive reals) and you get x = -5, the function is not onto for that y.

4. Wrong order in composition. (g ∘ f)(x) = g(f(x)), NOT f(g(x)). The right function (f) is applied first. Think of it as "g after f."

5. Forgetting to check all three properties for equivalence relations. If even one of reflexive, symmetric, or transitive fails, the relation is NOT an equivalence relation. Students sometimes prove two out of three and conclude "equivalence," losing marks for the missing property.

6. Not finding equivalence classes when asked. Board questions often end with "find the equivalence class of [2]" or "how many equivalence classes are there?" Students prove the relation is an equivalence relation but forget to answer the second part of the question. Read the full question before starting.

4-Week Practice Plan

Week 1: Relations

• Read NCERT Chapter 1 (Relations and Functions) — Sections 1.1 to 1.3
• Understand and memorize the definitions of all six types of relations
• Solve NCERT Examples 1.1 to 1.10 on relations
• Practice proving relations are reflexive, symmetric, transitive — at least 10 problems
• Make a one-page summary of all relation types with examples

Week 2: Functions — Types, Composition, and Inverse

• Study one-one, onto, and bijective functions — Sections 1.4 to 1.6
• Solve NCERT Examples 1.11 to 1.25
• Practice proving functions are one-one and onto — at least 10 problems
• Practice composition of functions — at least 10 problems
• Practice finding inverse functions — at least 10 problems
• Refer to the CBSE Class 12 Maths formulas list

Week 3: Binary Operations and Board Practice

• Study binary operations — Sections 1.7 to 1.8
• Practice finding identity and inverse elements — at least 8 problems
• Solve NCERT back exercises (Exercises 1.1 to 1.4, including Miscellaneous)
• Solve CBSE previous year questions from 2019-2026 — at least 25 questions from Chapter 1
• Time yourself: a 5-mark equivalence relation proof should take 7-8 minutes

Week 4: Revision and Mock Tests

• Revise the definitions of all relation types and function types daily
• Solve 2-3 complete Maths sample papers, focusing on the Relations and Functions questions
• Review common mistakes — especially the onto verification and composition order
• Practice equivalence relation proofs from scratch without looking at solutions
• Check the CBSE Class 12 Maths important chapters for the overall priority

Quick Revision Card for Exam Day

• Reflexive: (a,a) ∈ R for all a ∈ A
• Symmetric: (a,b) ∈ R → (b,a) ∈ R
• Transitive: (a,b) ∈ R and (b,c) ∈ R → (a,c) ∈ R
• Equivalence = Reflexive + Symmetric + Transitive (RST)
• One-one: f(x₁)=f(x₂) → x₁=x₂
• Onto: For all y in codomain, ∃ x in domain with f(x)=y
• Bijective = One-one + Onto → inverse exists
• Composition: (g∘f)(x) = g(f(x)) — apply f first, then g
• Inverse: y=f(x) → solve for x → f⁻¹(y)=x
• f⁻¹ ∘ f = Identity
• Binary operation: *: A×A→A; check closure, commutativity, associativity, identity, inverse
• Equivalence classes: [a] = {x ∈ A : xRa}; they partition the set

Start Practicing on Padhaao

Relations and Functions is a chapter that rewards understanding over memorization. The proofs follow a consistent structure — once you internalize the definitions and the three-step verification process for equivalence relations, this chapter becomes straightforward. Padhaao offers interactive concept explanations, guided proof practice, and board-style questions for every topic in Chapter 1. 10 learning modes, 3 free chapters daily, no download needed.

Start learning CBSE Class 12 Maths for free →

Frequently Asked Questions

How many marks does Relations and Functions carry in the board exam?

Chapter 1 carries approximately 5 marks out of 80 in the CBSE Class 12 Maths theory paper. The marks typically appear as one 5-mark question on proving a relation is an equivalence relation, or as a combination of a 3-mark question (prove one-one/onto) and a 2-mark question (composition or binary operation). Combined with Inverse Trigonometric Functions (Chapter 2), the unit contributes about 7 marks.

What is the most commonly asked 5-mark question?

"Show that the relation R on the set Z (or a given set A) defined by ... is an equivalence relation" is the most common 5-mark question. You must prove all three properties: reflexive, symmetric, and transitive. Some years, the question also asks you to find the equivalence class of a specific element, which adds 1-2 marks.

How do I prove a function is one-one?

Use the definition: assume f(x₁) = f(x₂) and then manipulate the equation algebraically to show x₁ = x₂. For example, if f(x) = 2x+5, then f(x₁) = f(x₂) gives 2x₁+5 = 2x₂+5, which simplifies to x₁ = x₂. Therefore f is one-one. For functions involving modulus or quadratics, you may need to use additional domain restrictions.

Is the binary operations section important?

Binary operations carry about 2 marks. Typical questions ask you to check if an operation is commutative or associative, or to find the identity/inverse element. These are direct definition-based questions and are easy marks if you understand the definitions. Do not skip this section — the marks-to-effort ratio is excellent.

How is this chapter different from Class 11 Relations and Functions?

Class 11 covers the basics: what a relation is, what a function is, domain, range, and types of functions (polynomial, rational, modulus). Class 12 goes deeper into the properties of relations (reflexive, symmetric, transitive, equivalence) and functions (one-one, onto, bijective), and introduces new concepts like composition, inverse functions, and binary operations. The focus shifts from "what is it" to "how to prove properties."

Do I need this chapter for JEE or NEET?

For JEE Main, this chapter is occasionally tested through questions on functions (one-one, onto, inverse) and composition. Equivalence relations appear rarely. For NEET, this chapter is not directly tested. However, the concepts of one-one and onto functions are foundational for higher mathematics, so understanding them deeply is beneficial regardless of which exam you are targeting.

Final Thoughts

Relations and Functions is the opening chapter of CBSE Class 12 Maths, and it introduces a style of mathematical thinking that many students find unfamiliar — proving properties systematically rather than computing answers. The good news is that once you learn the structure of an equivalence relation proof (reflexive → symmetric → transitive) and the technique for proving one-one/onto, every question in this chapter follows a predictable pattern. Master the definitions, practise the proof technique, and the 5 marks are yours. For the full preparation plan, see the CBSE Class 12 Maths chapter-wise marks weightage and the complete formulas list.