The Relations and Functions Class 11 Formulas list every formula, identity and theorem from Class 12 Mathematics Chapter 1 Relations and Functions in a single PDF. The Relations and Functions Class 11 Formulas are arranged section by section, with each formula shown beside the conditions under which it is valid. The formula sheet is downloadable as a free file aligned to the 2026-27 NCERT syllabus.
- CBSE Weightage: 4 to 6 marks (one short answer plus one MCQ or case-based item).
- JEE Main Weightage: 2 to 4% of the Maths section (1 to 2 questions per paper).
- CUET (UG) Weightage: 1 to 2 MCQs in nearly every shift.
This Formula Sheet is curated by Class 12 Maths experts, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.
All Formulas of Relations and Functions at a Glance
The Relations and Functions Class 11 Formulas address this in the same order as the NCERT textbook.
The learn table below catalogues every formula and counting result from Class 12 Maths Chapter 1, mapped to the exact NCERT section where it is introduced. Anchor your revision around these rows; each one has surfaced in a CBSE or JEE question in the last five years.
| Concept | Formula / Rule | NCERT Section | Common Use |
|---|---|---|---|
| Cartesian product cardinality | |A × B| = |A| · |B| | 1.1 | Counting ordered pairs |
| Number of relations from A to B | 2|A| · |B| | 1.2 | MCQ on relation counting |
| Number of relations on a set A | 2n2 where n = |A| | 1.2 | JEE counting questions |
| Reflexive relation | (a, a) ∈ R ∀ a ∈ A | 1.2 | RST property check (Step 1) |
| Symmetric relation | (a, b) ∈ R (b, a) ∈ R | 1.2 | RST property check (Step 2) |
| Transitive relation | (a, b), (b, c) ∈ R (a, c) ∈ R | 1.2 | RST property check (Step 3) |
| Equivalence relation | Reflexive AND Symmetric AND Transitive | 1.2 | 3-mark CBSE proof |
| Equivalence class of a | [a] = x ∈ A : (a, x) ∈ R | 1.2 | Listing classes |
| Number of reflexive relations on A | 2n2 - n | 1.2 (extension) | JEE Main |
| Number of symmetric relations on A | 2n(n+1)/2 | 1.2 (extension) | JEE Main |
| Empty relation | R = ⊆ A × A | 1.2 | Definition recall |
| Universal relation | R = A × A | 1.2 | Definition recall |
| Function (one-output rule) | f : A → B, ∀ a ∈ A, ∃ ! b ∈ B with f(a) = b | 1.3 | Distinguish from relation |
| One-one (injective) | f(x1) = f(x2) x1 = x2 | 1.3 | 2-mark CBSE proof |
| Onto (surjective) | ∀ y ∈ B, ∃ x ∈ A with f(x) = y | 1.3 | 2-mark CBSE proof |
| Bijective function | One-one AND Onto | 1.3 | Invertibility check |
| Number of functions from A to B | |B||A| | 1.3 (extension) | JEE counting |
| Number of one-one functions (m to n) | nPm = n!(n-m)! , with m ≤ n | 1.3 (extension) | JEE counting |
| Number of bijections from A to A | n! where n = |A| | 1.3 (extension) | JEE counting |
| Composition of functions | (g ∘ f)(x) = g(f(x)) | 1.4 (JEE extension) | JEE Main |
| Inverse of a bijection | f-1(y) = x f(x) = y | 1.4 (JEE extension) | JEE Main, inverse trig |
| Inverse of composition | (g ∘ f)-1 = f-1 ∘ g-1 | 1.4 (JEE extension) | JEE Main |
Around 6 of these 22 rules cover the entire 4 to 6 mark CBSE budget for the Relations and Functions Class 11 Formulas every year.
Relations and Functions Video Walkthrough
Source: Magnet Brains on YouTube
How the Relations and Functions Class 11 Formulas on the Relations and Functions Class 11 Formulas Help You
The Relations and Functions Class 11 Formulas address this in the same order as the NCERT textbook.
This sheet is built around the four habits CBSE toppers share when revising Chapter 1: RST template proofs, counting-rule recall, equivalence-class listing, and bijection checks.
- 2026-27 NCERT Alignment: Every formula matches the current edition, with JEE-only extensions (composition, inverse) clearly flagged so board-only students can skip them.
- Counting Formulas for MCQs: The number-of-relations, number-of-functions, and number-of-bijections rules covered explicitly, since they drive most 1-mark MCQs.
- Three-Step RST Template: The three rows for reflexive, symmetric, and transitive are written exactly in the order CBSE awards the 3 marks of an equivalence-relation proof.
- Quick-Revision Strip: The fact cards below collapse the 22-row table into 7 numerical recall items for the last 60 minutes before the exam.
Relations and Functions Quick-Fact Cards for MCQ Recall
The Relations and Functions Class 11 Formulas address this in the same order as the NCERT textbook.
The seven cards below are the highest-use numerical facts from Chapter 1. Memorise them as raw values; CBSE 1-mark MCQs and JEE counting questions almost always test one of these directly.
When to Use Which Formula in Relations and Functions
Picking the wrong rule loses you the entire mark of an MCQ even when the arithmetic is right. The decision tree below maps the typical question setup to the formula CBSE expects.
- Question asks "number of relations": use 2mn for relations from A to B, or 2n2 when the relation is on a single set.
- Question asks "number of functions": use |B||A| ; each element of A has |B| target choices, independent.
- Question asks "number of one-one functions": use nPm ; this fails when m > n (return zero in that case).
- Question asks "number of onto functions" from a small set: apply inclusion-exclusion: k=0n (-1)k nk (n-k)m . For the most asked Class 12 case |A| = |B|, the answer collapses to n!.
- Prove R is an equivalence relation: verify reflexive, symmetric, transitive in that order (3 marks, 1 each).
- List equivalence classes: pick a representative; sweep the set for everything related to it; repeat with an unused element until the set is exhausted.
- Check if f is bijective: run the one-one proof first (assume f(x1) = f(x2), derive x1 = x2), then the onto proof (start with arbitrary y, construct the pre-image).
Common Numerical Pattern Templates from CBSE and JEE
The four templates below summarise how Class 12 Chapter 1 questions are actually framed. Match the setup of the question on your paper to a template, then jump straight to the formula.
| Template | Typical Setup | Formula to Reach For |
|---|---|---|
| Counting relations | "How many relations are there from A to B where |A| = 3, |B| = 2?" | 2mn = 26 = 64 |
| Counting functions | "How many functions from a 4-element set to a 2-element set?" | |B||A| = 24 = 16 |
| Equivalence proof | "Show R = {(a, b) : 2 divides a - b} on Z is an equivalence relation." | RST 3-step template |
| Bijection check | "Check whether f(x) = 3 - 4x on R is bijective." | One-one + Onto proofs |
Memory Mnemonics for Relations and Functions
Use these short-form mnemonics in the last hour before the exam. They map the three abstract RST rules to concrete actions, which is what CBSE marking schemes actually reward.
One-Shot Revision Tips for Class 12th Maths Chapter 1
The five tips below summarise what to revise in the final 60 minutes before the board exam. Each one targets a specific high-frequency CBSE error pattern.
- Lock the RST order: always write the three checks in reflexive, symmetric, transitive sequence. Marking schemes award per-property; out-of-order proofs can lose half a mark.
- Test reflexivity with concrete values: for any rule like a + b = 10, check whether 2a = 10 admits a member of the set, not just whether the equation has a solution in general.
- Use numerical counterexamples for failure: CBSE rewards "let x = 1, y = 39, z = 20" over generic variables when a property fails.
- Range vs codomain: never declare a function onto without explicitly comparing its range to the stated codomain. This is the single most common 1-mark slip.
- Counting questions: read whether the question wants relations or functions; the formulas differ. Number of relations is 2mn; number of functions is |B||A|.
Top 5 Most-Asked Topics from Chapter 1 in Class 12 Board Exams
The short ranking below shows which sub-topics of Relations and Functions have driven the most marks in CBSE Class 12 Maths papers from 2021 to 2025. Anchor your last revision pass on these five.
- Equivalence-relation proof on a numeric rule (3 marks, almost every year).
- One-one and onto check for a linear function (2 marks, 4 of last 5 years).
- Counting relations or functions (1 mark MCQ, every year).
- Equivalence classes listing (2 marks, recurring).
- Identifying relation type from a given pair set (1 mark assertion-reason or MCQ).
Full year-wise PYQ map: Relations and Functions Class 12 Maths NCERT Solutions
More Relations and Functions Maths Class 12 Resources
- Relations and Functions Class 12 Maths NCERT Solutions
- Relations and Functions Class 12 Maths Notes
- Relations and Functions Class 12 Maths NCERT Book PDF
- Relations and Functions Class 12 Maths NCERT Exemplar Book PDF
- Relations and Functions Class 12 Maths NCERT Exemplar Solutions
- Relations and Functions Class 12 Maths Handwritten Notes
NCERT Formula Sheet for Class 12 Maths: All Chapters
Use the table below to jump to the Formula Sheet of any other Class 12 Maths chapter. Each link opens the same 22-row learn-table format you have on the Relations and Functions Class 11 Formulas.
| Chapter | Resource |
|---|---|
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
Relations and Functions Class 11 Formulas: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Relations and Functions Chapter
The Relations and Functions chapter splits into 2 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| 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 Relations and Functions Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| 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 |
The relations and functions class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- 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.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the relations and functions class 12 formulas pdf), and the full relations and functions class 12 pdf on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Relations and Functions Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Relations and Functions chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| 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. |
Walking through one example of each template before the exam covers most of the predictable relations and functions class 12 important questions you will see on board day.
- 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
The table below maps the dominant question type asked from the Relations and Functions chapter across recent CBSE Class 12 Maths boards:
| 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 |
The full relations and functions class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Relations and Functions Notes Pair with NCERT Solutions and the Formula Sheet
The Relations and Functions Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| 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 |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- 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
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| 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 recommended study plan for the Relations and Functions Class 12 chapter splits across three sittings. The table below outlines what to do in each:
| 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. |
For students preparing for both CBSE board and JEE Main:
- 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.
This Collegedunia NCERT Class 12 Mathematics page is reviewed against every CBSE board paper release.
The Collegedunia Class 12 Maths team checks every formula on this page against the printed NCERT 2026-27 edition.
Relations and Functions Class 11 Formulas - Frequently Asked Questions
Ques. Where can I download the Relations and Functions Class 11 Formulas for free?
Ans. You can download the Class 12 Maths Chapter 1 Relations and Functions Formula Sheet PDF directly from the download card on this page. Both the Normal and HD versions are free and cover every formula listed in the learn table above.
Ques. Is this Relations and Functions Class 11 Formulas aligned with the 2026-27 NCERT syllabus?
Ans. Yes. Every formula matches the current 2026-27 syllabus for Class 12 Maths. JEE-only extensions such as composition of functions and inverse of a bijection are clearly flagged as "extension" rows, so board-only students know they can skip them.
Ques. How many pages is the Class 12 Maths Chapter 1 Relations and Functions Formula Sheet PDF?
Ans. The Formula Sheet PDF runs 5 pages and packs every definition, counting rule, and equivalence-relation property from Chapter 1, plus the quick-fact card grid and the four CBSE-question templates.
Ques. What is the formula for the number of relations on a set in Class 12th Maths?
Ans. The number of relations on a set with n elements is 2n2. The number of relations from a set of size m to a set of size n is 2mn. Both are derived by counting subsets of the corresponding Cartesian product, which has n2 or mn ordered pairs.
Ques. How do I prove a relation is an equivalence relation in 3 marks?
Ans. Use the three-step RST template. Step 1 (1 mark): show (a, a) ∈ R for reflexivity. Step 2 (1 mark): assume (a, b) ∈ R and derive (b, a) ∈ R for symmetry. Step 3 (1 mark): assume (a, b) and (b, c) lie in R and derive (a, c) ∈ R for transitivity. Conclude by writing "Hence R is an equivalence relation."
Ques. How many bijections are there from a set of size n to itself?
Ans. Exactly n!. Each bijection is a permutation of the n elements, and the number of permutations of n distinct objects is n!. For n = 3, this gives 6 bijections; for n = 4, 24.
Ques. Are composition and inverse of functions in the CBSE Class 12 Maths 2026-27 syllabus?
Ans. Composition of functions and inverse of a bijection have been dropped from the CBSE Class 12 Maths board syllabus in the current 2026-27 edition, so they will not appear in your board paper. JEE Main, CUET, and most state CETs still test them, so JEE aspirants should keep the two extension rows of the learn table in their revision.
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