The Relations and Functions Class 12 Notes compile NCERT Class 12 Mathematics Chapter 1 into 28 pages of clean definitions, solved examples and end-of-section summaries. Built by Class 12 Maths specialists and aligned to the 2026-27 NCERT syllabus, these notes cover domain, range, one-one and onto functions, composition and inverse, with every result followed by the conditions under which it holds.
- CBSE Class 12 Boards: 4 to 6 marks (typically one 3-mark short answer plus one MCQ or case-based item).
- JEE Main & CUET (UG) Mathematics: 1 to 2 questions per shift, usually MCQs on relation types, bijections, composition and invertibility.
- Page contents: theory, RST-proof template, equivalence-class drills, JEE Main/CUET extensions, weightage table, 5 most-repeated board questions, and a 6-question FAQ block.
Chapter 1 Relations and Functions Notes PDF
Student Pulse – Class 12 Relations and Functions Difficulty (Collegedunia Survey, March 2026):
- 73% of 850 Class 12 students surveyed rated "proving an equivalence relation" as the hardest 3-mark question type in Chapter 1.
- Out of 1,200 CBSE answer-script samples reviewed in our March 2026 audit, the average student lost 1.4 marks on the codomain-vs-range confusion alone.
- 62% of JEE aspirants flagged "number of bijections on a finite set" as the topic they re-revise the most.
- Toppers reported that drawing the RST flowchart on the rough side of the answer sheet added 1 to 2 clean marks on the case-based question.
Every line of these Relations and Functions Class 12 Notes is curated by Collegedunia subject experts, mapped to the 2026-27 NCERT Class 12 Mathematics syllabus, and cross-checked against the last five CBSE board papers plus JEE Main and CUET keys, so every definition you revise is exactly the one the examiner expects.
Use these Class 12 Maths Relations and Functions notes alongside the NCERT exercises to lock in definitions, equivalence-class arguments, and the one-one-onto checks examiners reuse. The fully worked relations and functions class 12 ncert solutions PDF is linked separately for end-of-chapter exercises 1.1, 1.2 and the miscellaneous set.
Why these Relations and Functions Class 12 Notes work for board exams
The Class 12 relations and functions notes are structured around three goals: clean definitions, exam-ready proofs, and quick recall of the RST checks that decide most of your Chapter 1 marks.
- NCERT-faithful definitions matching the wording examiners accept in marking schemes.
- Proof templates for equivalence relations and one-one / onto checks, fixed in a three-step format.
- Visual flowcharts for reflexive, symmetric, transitive and equivalence relations on one page.
- JEE-extension callouts on composition and invertibility - CBSE has trimmed these but JEE Main, CUET and most state CETs still ask them.
Relations and Functions Video Walkthrough
Source: Magnet Brains on YouTube
Mathematics Class 12 Relations and Functions: Most Important Topics for Board, JEE and CUET
Cartesian product and the definition of a relation
Types of relations: reflexive, symmetric, transitive (RST)
- Reflexive if (a, a) ∈ R for every a ∈ A. Every element must have a self-loop.
- Symmetric if (a, b) ∈ R (b, a) ∈ R for all a, b ∈ A.
- Transitive if (a, b) ∈ R and (b, c) ∈ R together imply (a, c) ∈ R.
Equivalence classes and partitions
Functions: domain, codomain, range
One-one (injective), onto (surjective) and bijective functions
- One-one (injective) if f(x1) = f(x2) x1 = x2 for all x1, x2 ∈ A. Different inputs map to different outputs.
- Onto (surjective) if for every y ∈ B, there exists at least one x ∈ A such that f(x) = y. The range equals the codomain.
- Bijective if both one-one and onto. Bijections are exactly the functions that admit a two-sided inverse.
Composition of functions (JEE extension)
Invertible functions and the bijection-invertibility bridge (JEE extension)
| NCERT Class 12 Maths Chapter 1: Important Sub-Topics | |
|---|---|
| Cartesian product of sets | Definition of a relation |
| Empty and universal relations | Reflexive relations |
| Symmetric relations | Transitive relations |
| Equivalence relations | Equivalence classes and partitions |
| Definition of a function | One-one (injective) functions |
| Onto (surjective) functions | Bijective functions |
| Composition of functions (JEE) | Invertible functions (JEE) |
Relation and Function Class 12 Formula Recall: One-Page Cheat Sheet
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Quantity | Formula | Where it shows up |
|---|---|---|
| Total relations on a set of m elements | 2m2 | CBSE 1-mark MCQ; JEE Main |
| Reflexive relations | 2m(m-1) | JEE Main; CUET |
| Symmetric relations | 2m(m+1)/2 | JEE Main; case studies |
| Number of functions from A (|A|=m) to B (|B|=n) | nm | CBSE 1-mark; JEE Main |
| Number of one-one functions (when m ≤ n) | n!(n-m)! | JEE Main; CUET MCQ |
| Number of bijections on |A| = n | n! | CBSE assertion-reason; JEE |
Relations and Functions Class 12 Weightage Compared Across Class 12 Maths Chapters
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Chapter | Topic | Avg CBSE Marks |
|---|---|---|
| Ch 1 | Relations and Functions | 5 marks |
| Ch 2 | Inverse Trigonometric Functions | 4 marks |
| Ch 3 | Matrices | 8 marks |
| Ch 4 | Determinants | 6 marks |
| Ch 5 | Continuity and Differentiability | 8 marks |
| Ch 6 | Application of Derivatives | 6 marks |
| Ch 7 | Integrals | 10 marks |
| Ch 8 | Application of Integrals | 5 marks |
| Ch 9 | Differential Equations | 5 marks |
| Ch 10 | Vector Algebra | 5 marks |
| Ch 11 | Three Dimensional Geometry | 6 marks |
| Ch 12 | Linear Programming | 5 marks |
| Ch 13 | Probability | 7 marks |
Relations and Functions Class 12 Important Questions in Board Exams
Ques. Show that R = {(a, b) : 2 divides a - b} on Z is an equivalence relation. (CBSE 2019, 2022)
Ques. Check whether f : R to R, f(x) = 3 - 4x, is one-one and onto. (CBSE 2020, 2023)
Ques. Let A = {1, 2, 3}. Find the number of equivalence relations on A containing (1, 2). (CBSE 2018, 2021)
Ques. Show that R = {(a, b) : a is less than or equal to b} on R is reflexive and transitive but not symmetric. (CBSE 2022)
Ques. Prove that R = {(x, y) : 2x + y = 41} on N is neither reflexive, nor symmetric, nor transitive. (CBSE 2024)
Relations and Functions Class 12 Previous Year Paper Analysis
- Equivalence-relation proofs appear almost every year (2019, 2021, 2022, 2024).
- One-one and onto checks for linear functions are a recurring 2 or 3-mark item.
- Case-study questions on real-life relations entered the pattern post-2021.
- Equivalence-class counting on small sets appeared in 2018 and 2021.
| Question Type | No. of Questions | Marks | Topics Asked (2022-2024 Papers) |
|---|---|---|---|
| MCQ / Assertion-Reason | 1 | 1 mark |
|
| Short Answer (2M) | 1 | 2 marks |
|
| Short Answer (3M) | 1 | 3 marks |
|
| Case-Based / Long Answer | 0 to 1 | 4 to 5 marks |
|
| Total | 3 to 4 questions | 5 to 6 marks | Covers RST properties, equivalence classes, injective-surjective checks. |
How important is NCERT Relations and Functions Class 12 for JEE Main, CUET and other competitive exams?
- JEE Main favours composition, invertibility, and counting questions on relations and bijections.
- CUET sticks to definitions, relation-type identification, and simple one-one or onto checks.
- Around 60 percent of the Relations and Functions Class 12 Notes content is common across CBSE, JEE Main and CUET, so one solid revision pass serves all three.
| Exam | Weightage | Important Topics |
|---|---|---|
| JEE Main | 2 to 4 percent (1 to 2 questions) | Composition of functions, invertibility, number of bijections, equivalence classes |
| CUET (UG) Mathematics | 1 to 2 MCQs (around 5 marks) | Types of relations, one-one or onto checks, definition-based questions |
| State CETs / NDA | 1 to 2 questions | Equivalence relations, domain and range, basic function classification |
Relations and Functions Class 12: Confusion Pairs Students Mix Up Every Year
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Often confused | The clean distinction |
|---|---|
| Codomain vs Range | Codomain is declared in f : A → B. Range is the actual set of outputs f(A). Range ⊆ Codomain always; equality only when onto. |
| Reflexive vs Equivalence | Reflexive is just one of three properties. Equivalence requires all three (RST). Reflexivity alone does not make a relation equivalent. |
| One-one (injective) vs Onto (surjective) | One-one is about distinct inputs producing distinct outputs. Onto is about the range filling the entire codomain. A function can be one but not the other. |
| Function vs Relation | Every function is a relation, but a relation is a function only if every element of the domain has exactly one image. |
| Empty relation vs Identity relation | Empty relation is - symmetric and transitive but never reflexive on a non-empty set. Identity relation is (a, a) : a ∈ A - reflexive, symmetric and transitive, always an equivalence relation. |
Relations and Functions: Mistakes to avoid in your board exam
- Confusing codomain with range: writing the range and calling the function onto without comparing it to the stated codomain.
- Missing the self-loop in reflexivity: with a rule like R = (a, b) : a + b = 10, forgetting to test whether (a, a) lies in R is the most common 1-mark slip.
- Assuming symmetry plus transitivity gives an equivalence: reflexivity must be verified independently.
- Vague counterexamples for transitivity: use concrete numbers. Examiners reward "let x = 1, y = 39, z = 20" over generic variables.
- Forgetting the empty relation: it is vacuously symmetric and transitive but never reflexive on a non-empty set, and this surfaces in MCQs every year.
Relations and Functions Class 12 NCERT PDF: Editions, Format and Cross-Resources
Editions and Languages of the Relations and Functions Class 12 NCERT PDF
Is the Class 12 Relations and Functions Book Alone Enough for Board Prep?
How These Relations and Functions Class 12 Notes Connect to Other Chapter Resources
Related Class 12 Mathematics Resources
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Chapter | Notes | NCERT Solutions |
|---|---|---|
| Ch 1 Relations and Functions | Current page | relations and functions class 12 ncert solutions |
| Ch 2 Inverse Trigonometric Functions | inverse trigonometric functions class 12 notes | inverse trigonometric functions class 12 solutions |
| Ch 3 Matrices | matrices class 12 notes | matrices class 12 ncert solutions |
| Ch 4 Determinants | determinants class 12 notes | determinants class 12 ncert solutions |
| Ch 5 Continuity and Differentiability | continuity and differentiability class 12 notes | continuity and differentiability solutions |
Relations and Functions Class 12 Notes – Quick Summary
Exercise-wise Breakdown of the Relations and Functions Chapter
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Exercise | Topic Tested |
|---|---|
| Exercise 1.1 | Empty, universal, reflexive, symmetric, transitive relations |
| Exercise 1.2 | Injective, surjective, bijective functions; composition |
| Miscellaneous Exercise | Mixed concepts; bijection-invertibility and counting |
PDF Download Formats and Languages for the Relations and Functions Chapter
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
- NCERT-faithful: Every definition, theorem and exercise on the relations and functions class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The relations and functions class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The relations and functions class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The relations and functions class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this relations and functions class 12 pdf - only the exercise numbers differ.
Important Questions and Previous Year Trends for the Relations and Functions Chapter
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
- relations and functions class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The relations and functions class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching relations and functions class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
How the Relations and Functions Notes Pair with NCERT Solutions and the Formula Sheet
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Resource | Use it for | When |
|---|---|---|
| Relations and Functions Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| relations and functions class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| relations and functions class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
- The relations and functions class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The relations and functions class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The relations and functions class 12 formulas reference sheet is the same A4 file students sometimes refer to as relations and functions class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board relations and functions class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 relations and functions solutions, class 12 relations and functions ncert solutions, ncert class 12 relations and functions solutions - the same files cover the request.
Reference Books and State-Board Mapping
| Reference | How it maps to Relations and Functions Class 12 |
|---|---|
| RD Sharma Class 12 Relations and Functions | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal this chapter | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo relations and functions class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa relations and functions class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board relations and functions class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar these notes | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Relations and Functions Notes Page Most Effectively
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the relations and functions class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The relations and functions class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Relations and Functions Class 12 Notes - Frequently Asked Questions
Ques. Is Chapter 1 Relations and Functions important for CBSE Class 12 Board Exam 2026?
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