Complete Summary and Solutions for Relations and Functions – NCERT Class XII Mathematics Part I, Chapter 1 – Types of Relations, Functions, Domain, Range, Types of Functions, Composition, Inverse Functions

Comprehensive summary and explanation of Chapter 1 'Relations and Functions' from NCERT Class XII Mathematics Part I textbook, covering ordered pairs, Cartesian product, different types of relations, types of functions including one-one, onto, bijection, composition and inverse of functions, with examples, properties, and all NCERT exercises and solutions.

Updated: 7 months ago

Categories: NCERT, Class XII, Mathematics Part I, Chapter 1, Relations, Functions, Domain, Range, Composition, Inverse, Summary, Questions, Answers

Tags: Relations, Functions, Domain, Range, One-one Function, Onto Function, Bijection, Composition of Functions, Inverse Functions, NCERT, Class 12, Mathematics, Summary, Explanation, Questions, Answers, Chapter 1

Relations and Functions - Class 12 Mathematics Chapter 1 Ultimate Study Guide 2025

Full Chapter Summary & Detailed Notes - Relations and Functions Class 12 NCERT

Overview & Key Concepts

Chapter Goal: Build on Class XI; focus on types of relations (reflexive, symmetric, transitive, equivalence), functions (one-one, onto, bijective), composition, invertible. Exam Focus: Definitions (3-7), Examples (1-26), Equivalence classes, Finite vs Infinite sets. Fun Fact: Equivalence partitions sets. Core Idea: Relations as subsets; functions as special relations. Real-World: Congruence in geometry. Expanded: All subtopics point-wise with evidence (e.g., Ex 2 congruence), examples (e.g., even/odd integers), debates (finite bijection vs infinite).
Wider Scope: From relations to binary ops tease; sources: Pages 1-17, Figs 1.1-1.5.
Expanded Content: Include composition/invertible; point-wise for recall; add 2025 relevance like graph theory apps.

1.1 Introduction

Relations: Subset of \( A \times B \); \( aRb \) if \( (a,b) \in R \). Ex: Students A to B (brother/sister, age, marks).
Functions: Special relations (Class XI review: domain, codomain, range).
Objectives: Types of relations/functions, composition, invertible, binary ops.
Expanded: Evidence: English analogy; debates: Abstract vs real links.

Conceptual Diagram: Relation vs Function

Relation: Arrows from A to B (any pairs). Function: Exactly one arrow per A element. Ties to Fig 1.2.

Why This Guide Stands Out

Comprehensive: All subtopics point-wise, solved examples integrated; 2025 with proofs, processes analyzed for injectivity/surjectivity.

1.2 Types of Relations

Empty/Universal: \( R = \phi \), \( R = A \times A \). Ex 1: Boys school sister (empty), height diff <3m (universal).
Reflexive/Symmetric/Transitive: Def 3 (i-iii). Symmetric: \( aRb \implies bRa \).
Equivalence: All three (Def 4). Ex 2: Congruent triangles (Fig 1.1). Ex 3: Perpendicular lines (symmetric, not ref/trans). Ex 4: {1,2,3} partial. Ex 5: Even diff in Z.
Equivalence Classes: Partitions [a] = {b | bRa}. Ex: Even/odd [0],[1]; mod 3 [0],[1],[2]. Ex 6: Odd/even in {1-7}.
Expanded: Evidence: Partitions disjoint/union A; reverse: Subsets define R.

Quick Table: Types of Relations

Type	Definition	Example
Empty	No pairs	Sister in boys school
Universal	All pairs	Height diff <3m
Reflexive	\( (a,a) \in R \)	Even diff
Symmetric	\( aRb \implies bRa \)	Perpendicular
Transitive	\( aRb, bRc \implies aRc \)	Congruent
Equivalence	All three	Mod 3 classes

1.3 Types of Functions

One-One (Injective): Def 5; distinct images. Ex 7: Roll no. (one-one, not onto). Ex 8: \( f(x)=2x \) N→N.
Onto (Surjective): Def 6; every y image. Remark: Range=Y.
Bijective: Both (Def 7). Ex 9: \( f(x)=2x \) R→R. Ex 10: Piecewise N→N onto not one-one. Ex 11: \( x^2 \) neither. Ex 12: Odd/even shift bijective.
Finite Sets: One-one iff onto (Ex 13-14). Infinite: Not (Ex 8,10).
Expanded: Evidence: Diagrams Fig 1.2; debates: Infinite counterexamples.

1.4 Composition & Invertible

Composition: Def 8 \( (g \circ f)(x) = g(f(x)) \). Ex 15: Sets {2,3,4,5}→{7,11}. Ex 16: cos x & 3x² ≠ commute (Fig 1.5).
Invertible: Def 9; \( g \circ f = I_X, f \circ g = I_Y \). Bijection iff invertible. Ex 17: \( f(x)=4x+3 \) N→Y; inverse \( g(y)=(y-3)/4 \).
Expanded: Evidence: Proves bijection; real: Encryption functions.

Miscellaneous Examples

Ex 18: Intersection equivalence. Ex 19: Ordered pairs xv=yu. Ex 20: Mod 3 same as subsets. Ex 21: Kernel equivalence. Ex 22: 3! one-one. Ex 23: 3 relations reflexive/trans not sym. Ex 24: 2 equiv containing (1,2)(2,1). Ex 25: IN onto, 2IN not. Ex 26: sin+cos not one-one.
Expanded: Evidence: Proofs; debates: Finite permutations.

Summary & Exercises

Key Takeaways: Relations classify; functions map uniquely; bijections invert.
Exercises Tease: Check properties; prove bijections.

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance. Expanded: 20+ terms grouped by subtopic; added advanced like "Equivalence Class", "Composition" for depth/easy flashcards. Use MathJax for precision.

Relation

Subset of \( A \times B \); \( aRb \) if \( (a,b) \in R \). Ex: Age comparison. Relevance: Links sets.

Empty Relation

\( R = \phi \). Ex: Sister in boys school. Relevance: Trivial case.

Universal Relation

\( R = A \times A \). Ex: All pairs. Relevance: Full connection.

Reflexive

\( (a,a) \in R \ \forall a \in A \). Ex: \( |a-b| \geq 0 \). Relevance: Self-relation.

Symmetric

\( aRb \implies bRa \). Ex: Congruent. Relevance: Bidirectional.

Transitive

\( aRb, bRc \implies aRc \). Ex: Even diff. Relevance: Chain.

Equivalence Relation

Ref + Sym + Trans. Ex: Modulo. Relevance: Partitions.

Equivalence Class

\( [a] = \{ b \mid bRa \} \). Ex: [0] evens. Relevance: Subsets.

One-One (Injective)

\( f(x_1)=f(x_2) \implies x_1=x_2 \). Ex: \( 2x \). Relevance: Unique inputs.

Onto (Surjective)

Every y has x with f(x)=y. Ex: Range=Y. Relevance: Covers codomain.

Bijective

Injective + Surjective. Ex: Identity. Relevance: Invertible.

Composition

\( (g \circ f)(x) = g(f(x)) \). Ex: g(cos x). Relevance: Chain functions.

Invertible Function

\( f^{-1} \) exists; bijection. Ex: \( g(y)=(y-3)/4 \). Relevance: Reverses.

Domain/Codomain/Range

Domain: Inputs; Codomain: Possible outputs; Range: Actual. Ex: f: N→N, range evens. Relevance: Function spec.

Tip: Group by relations/functions; examples for recall. Depth: Debates (e.g., finite bijection). Historical: Dirichlet. Interlinks: To Ch2 Inverse Trig. Advanced: Binary ops. Real-Life: Database relations. Graphs: Fig 1.2-1.5. Coherent: Evidence → Interpretation. For easy learning: Flashcard per term with Ex snippet.

Solved Examples from NCERT - Step-by-Step with MathJax

All 26 examples solved; grouped by section. Expanded with steps, diagrams desc.

Example 1: Empty/Universal in Boys School

Solution: R sister: No pairs → empty. Height diff <3m: All pairs → universal.

Example 2: Congruent Triangles Equivalence

Solution: Reflexive: T ≅ T. Symmetric: T1 ≅ T2 ⇒ T2 ≅ T1. Transitive: Chain. Thus equivalence.

\[ R = \{(T_1, T_2) \mid T_1 \cong T_2\} \]

Example 3: Perpendicular Lines (Fig 1.1)

Solution: Not reflexive (L ⊥ L false). Symmetric: Yes. Not transitive (L1 ⊥ L2, L2 ⊥ L3 ⇒ L1 // L3).

Example 4: Partial in {1,2,3}

Solution: Reflexive: Diagonals. Not sym: (1,2) no (2,1). Not trans: (1,2),(2,3) no (1,3).

\[ R = \{(1,1),(2,2),(3,3),(1,2),(2,3)\} \]

Example 5: Even Diff in Z

Solution: Reflexive: 2|a-a|. Sym: 2|a-b| ⇒ 2|b-a|. Trans: a-c = (a-b)+(b-c) even. Equivalence; [0] evens, [1] odds.

\[ R = \{(a,b) \mid 2 \mid a-b\} \]

Example 6: Odd/Even in {1-7}

Solution: Ref/Sym/Trans: All odd/even same parity. Classes: {1,3,5,7}, {2,4,6}.

Example 7: Roll Numbers One-One Not Onto

Solution: Distinct rolls → one-one. 51 not image → not onto.

Example 9: \( f(x)=2x \) R→R Bijective

Solution: One-one: 2x1=2x2 ⇒ x1=x2. Onto: y=2(x=y/2). (Fig 1.3)

\[ f(x_1) = f(x_2) \implies x_1 = x_2; \quad y = 2 \left( \frac{y}{2} \right) \]

Example 11: \( x^2 \) Neither

Solution: f(-1)=f(1)=1 not one-one. -2 no preimage (Fig 1.4).

Example 12: Piecewise Bijective

Solution: One-one: Cases separate. Onto: Odds from evens+1, evens from odds-1.

\[ f(x) = \begin{cases} x+1 & x \ odd \\ x-1 & x \ even \end{cases} \]

Example 13: Finite Onto ⇒ One-One

Solution: Contradict: Not one-one ⇒ range <3, not onto.

Example 15: Composition gof

Solution: gof(2)=7, gof(3)=7, gof(4)=11, gof(5)=11.

Example 16: Non-Commute

Solution: \( g(f(x)) = 3\cos^2 x \neq \cos(3x^2) = f(g(x)) \) at x=0.

Example 17: Invertible \( f(x)=4x+3 \)

Solution: Bijection; \( g(y) = \frac{y-3}{4} \); gof=IN, fog=IY.

Example 18: Intersection Equivalence

Solution: Ref/Sym/Trans inherited.

Example 19: Ordered Pairs xv=yu

Solution: Ref: xy=yx. Sym: Swap. Trans: Chain multiply.

Example 20: Mod 3 = Subsets

Solution: Diff multiple 3 iff same subset {1,4,7} etc.

Example 21: Kernel Equivalence

Solution: Ref/Sym/Trans via f(a)=f(b).

Example 22: One-One Functions {1,2,3}→Self

Solution: 3! = 6 permutations.

Example 23: Relations Ref/Trans Not Sym

Solution: Base + (2,1) or + (3,2); 3 total.

Example 24: Equiv Containing (1,2)(2,1)

Solution: Minimal + universal; 2.

Example 25: IN Onto, 2IN Not

Solution: 3 no preimage under 2x.

Example 26: sin+cos Not One-One

Solution: f(0)=1=f(π/2).

Tip: Steps: Def → Check → Conclude. Practice proofs.

Exercise 1.1 Questions & Answers - With MathJax Solutions

16 Qs from book; solutions step-by-step. Grouped short/long.

1(i): R in A={1..14}, 3x-y=0

Solution:

Ref: y=3x, x=3x? No. Not ref. Sym: y=3x ⇒ x=3y? No. Not sym. Trans: Possible chain no. Not trans.

1(ii): N, y=x+5, x<4

Solution: Pairs (1,6),(2,7),(3,8). Not ref (no (a,a)). Not sym ((1,6) no (6,1)). Not trans ((1,6),(6,11)? No pair).

2: R={(a,b): a ≤ b²} Neither

Solution: Not ref: a=2, 2≤4 yes but a=-2 no. Not sym: 2≤1? No but 1≤4. Not trans: 1≤1,1≤4 but 1≤16? Chain issue.

3: {1-6}, b=a+1

Solution: Not ref (no (a,a)). Not sym ((1,2) no (2,1)). Not trans ((1,2),(2,3) ⇒ (1,3) yes, but chain limited).

\[ R = \{(1,2),(2,3),(3,4),(4,5),(5,6)\} \]

4: ≤ Reflexive Trans Not Sym

Solution: Ref: a≤a. Trans: a≤b≤c ⇒ a≤c. Not sym: 1≤2 no 2≤1.

5: ≤ b³

Solution: Ref: a≤a³? For a=2 no. Not ref. Sym: 1≤8 yes,8≤1 no. Not sym. Trans: Possible no.

6: {(1,2),(2,1)} Sym Not Ref/Trans

Solution: Sym: Yes swap. Not ref: No (1,1). Not trans: No chain.

7: Books Same Pages Equiv

Solution: Ref: Same as self. Sym: Mutual. Trans: Chain same. Equiv.

8: |a-b| Even in {1-5} Equiv

Solution: Ref: 0 even. Sym/Trans: Even diff. Classes: Odds {1,3,5}, Evens {2,4}.

9(i): |a-b| Multiple 4 in {0-12}

Solution: Equiv (like mod 4). [1]={1,5,9}.

9(ii): a=b Identity Equiv

Solution: Trivial equiv. [1]={1}.

10: Examples

Solution: (i) Sym not ref/trans: {(1,2),(2,1)}. (ii) Trans not ref/sym: ≤ on N. (iii) Ref sym not trans: Equal size subsets. (iv) Ref trans not sym: Precedes. (v) Sym trans not ref: Empty.

11: Distance from Origin Equiv

Solution: Ref/sym/trans: Same dist. [P]: Circle center O radius OP.

12: Similar Triangles Equiv

Solution: Ref/sym/trans angles/sides. T1~T3 (3-4-5 scale 2), not T2.

13: Same Sides Polygons Equiv

Solution: Ref/sym/trans count. [T]: All triangles.

14: Parallel Lines Equiv

Solution: Ref (// self), sym, trans (transitive). [y=2x+4]: All slope 2.

15: {1,2,3} R={(1,2),(2,3),(1,3),(3,1),(2,1),(3,2),(1,1),(2,2),(3,3)} Equiv

Solution: Full; (D) Equiv.

16: a=b-2, b>6 None

Solution: Pairs like (6,8) no (a=6,b=8:8=6-2? No). (D) (8,7):7=8-2 no.

Tip: Check each property systematically.

Exercise 1.2 Questions & Answers - With MathJax

12 Qs; injectivity/surjectivity proofs.

1: \( f(x)=1/x \) R→R Bijection

Solution: One-one: 1/x1=1/x2 ⇒ x1=x2. Onto: y=1/(1/y). N domain: One-one not onto (1 no preimage).

\[ f(x_1)=f(x_2) \implies x_1=x_2; \quad f(1/y)=y \]

2(i): N→N x² One-One Not Onto

Solution: x1²=x2² ⇒ |x1|=|x2| but N positive ⇒ x1=x2. Not onto: 2 no square root integer.

2(ii): Z→Z x² Not One-One Onto? No

Solution: (-1)²=1² not one-one. Onto? No, -2 no.

2(iii): R→R x² Not One-One Not Onto

Solution: Same; negatives no image.

2(iv): N→N x³ One-One Not Onto

Solution: Strict increase. 2 no cube root.

2(v): Z→Z x³ Bijection

Solution: Strict; every integer cube root exists? Wait, cubes cover all Z? Yes, odd function bijective.

3: [x] Greatest Integer Neither

Solution: [1.5]=[1.6]=1 not one-one. Not onto: 1.5 no integer x.

4: |x| Modulus Neither

Solution: |-1|=|1| not one-one. Negatives no (range [0,∞)).

5: Signum Neither

Solution: sgn(1)=sgn(2)=1 not one-one. -1 image but 0 only at 0.

\[ \sgn(x) = \begin{cases} 1 & x>0 \\ 0 & x=0 \\ -1 & x<0 \end{cases} \]

6: f={(1,4),(2,5),(3,6)} One-One

Solution: Distinct images.

7(i): 3-4x Bijection

Solution: Linear strict; inverse exists.

7(ii): 1+x² Not One-One Onto? No

Solution: x²=(-x)²; range [1,∞) not all R.

8: f(a,b)=(b,a) Bijection

Solution: Inverse self; one-one/onto obvious.

9: Piecewise Odd/Even Bijection

Solution: Like Ex12; covers all.

10: \( f(x) = \frac{2x}{x-3} \) A=R-{3}→B=R-{1} Bijection

Solution: One-one: Solve eq. Onto: y(x-3)=2x ⇒ x=3y/(y-2) ≠1.

11: x⁴ Neither

Solution: (D) Neither (even power).

12: 3x Bijection

Solution: (A) One-one onto.

Tip: Prove: Assume eq, solve; for onto solve f(x)=y.

Subtopic	Key Points	Examples	Mnemonics/Tips
Relations Types	Ref/Sym/Trans: Self/Bidir/Chain. Equiv: All three → partitions.	Congruent; even diff.	RST-E (Reflex Sym Trans Equiv). Tip: "Relations: Reflect, Swap, Transit to Equiv Partition".
Functions Types	One-One: Unique img. Onto: Full cov. Biject: Both; finite iff.	2x R biject; N not onto.	IOB-F (Inject Onto Biject Finite). Tip: "Inject Unique, Onto Covers, Biject Inverts; Finite Special".
Composition/Invert	g∘f: g(f(x)). Inv: Bijection; f⁻¹.	cos(3x²) ≠; 1/x inv.	CI (Compose Inside; Invert Bij). Tip: "Compose Chain, Invert Reverse Only Bij".
Equiv Classes	[a]: All related to a; disjoint union A.	[0] evens mod 2.	DU (Disjoint Union). Tip: "Classes: Divide And Conquer Set".

This article has been published in CBSE Class 12 Board Examination. Explore everything related to it here.

Recommended BooksSee All →

35% OFF

Bhagvad Gita – The Story Way for Students & Parents

₹194M.R.P. ₹299

Buy on Amazon

34% OFF

You Can – The Classic Guide to Self-Belief & Personal Growth

₹99M.R.P. ₹149

Buy on Amazon

Atomic Habits – Tiny Changes, Remarkable Results

₹521

Buy on Amazon

As an Amazon Associate, ProSyllabus earns from qualifying purchases. Prices shown are subject to change.

Practice Quiz

Test your CBSE Class 12 Board Examination prep

🎁New here? Your first purchase is just ₹1 — 120 coins with code RAMANUJAN_1

Quizzes

10 questions · ~10 minutes · instant rank & AI diagnosis

5 attempts on CBSE Class 12 Board Examination quizzes

How scored 1/10 on Economics (Class 12) Ravi jatav scored 0/10 on Economics (Class 12) Ravi jatav scored 0/10 on Economics (Class 12) Ravi jatav scored 0/10 on Accountancy (Class 12) Dhara scored 3/10 on Electrostatic Potential and Capacitance How scored 1/10 on Economics (Class 12) Ravi jatav scored 0/10 on Economics (Class 12) Ravi jatav scored 0/10 on Economics (Class 12) Ravi jatav scored 0/10 on Accountancy (Class 12) Dhara scored 3/10 on Electrostatic Potential and Capacitance

Economics (Class 12) Practice Quiz | CBSE Class 12 Board Examination

10 Qs · ~10 min

Accountancy (Class 12) Practice Quiz | CBSE Class 12 Board Examination

10 Qs · ~10 min

Electrostatic Potential and Capacitance Fundamentals | CBSE Class 12 Board Examination

10 Qs · ~10 min

Class 12 English — Memories of Childhood (Practice Quiz)

10 Qs · ~10 min

Class 12 English — On the Face of It (Practice Quiz)

10 Qs · ~10 min

Class 12 English — The Enemy (Practice Quiz)

10 Qs · ~10 min

Class 12 English — Journey to the End of the Earth (Practice Quiz)

10 Qs · ~10 min

Class 12 English — The Tiger King (Practice Quiz)

10 Qs · ~10 min

Class 12 English — The Third Level (Practice Quiz)

10 Qs · ~10 min

#10

Class 12 English — Aunt Jennifer's Tigers (Practice Quiz)

10 Qs · ~10 min

#11

Class 12 English — A Roadside Stand (Practice Quiz)

10 Qs · ~10 min

#12

Class 12 English — A Thing of Beauty (Practice Quiz)

10 Qs · ~10 min

#13

Class 12 English — Keeping Quiet (Practice Quiz)

10 Qs · ~10 min

#14

Class 12 English — An Elementary School Classroom in a Slum (Practice Quiz)

10 Qs · ~10 min

#15

Class 12 English — My Mother at Sixty-six (Practice Quiz)

10 Qs · ~10 min

#16

Class 12 English — Going Places (Practice Quiz)

10 Qs · ~10 min

#17

Class 12 English — The Interview (Practice Quiz)

10 Qs · ~10 min

#18

Class 12 English — Poets and Pancakes (Practice Quiz)

10 Qs · ~10 min

#19

Class 12 English — Indigo (Practice Quiz)

10 Qs · ~10 min

#20

Class 12 English — The Rattrap (Practice Quiz)

10 Qs · ~10 min

#21

Class 12 English — Deep Water (Practice Quiz)

10 Qs · ~10 min

#22

Class 12 English — Lost Spring (Practice Quiz)

10 Qs · ~10 min

#23

Class 12 English — The Last Lesson (Practice Quiz)

10 Qs · ~10 min

#24

Class 12 Psychology — Social Influence and Group Processes (Practice Quiz)

10 Qs · ~10 min

#25

Class 12 Psychology — Attitude and Social Cognition (Practice Quiz)

10 Qs · ~10 min

#26

Class 12 Psychology — Therapeutic Approaches (Practice Quiz)

10 Qs · ~10 min

#27

Class 12 Psychology — Psychological Disorders (Practice Quiz)

10 Qs · ~10 min

#28

Class 12 Psychology — Meeting Life Challenges (Practice Quiz)

10 Qs · ~10 min

#29

Class 12 Psychology — Self and Personality (Practice Quiz)

10 Qs · ~10 min

#30

Class 12 Psychology — Variations in Psychological Attributes (Practice Quiz)

10 Qs · ~10 min

#31

Class 12 Sociology — Social Movements (Practice Quiz)

10 Qs · ~10 min

#32

Class 12 Sociology — Mass Media and Communications (Practice Quiz)

10 Qs · ~10 min

#33

Class 12 Sociology — Globalisation and Social Change (Practice Quiz)

10 Qs · ~10 min

#34

Class 12 Sociology — Change and Development in Industrial Society (Practice Quiz)

10 Qs · ~10 min

#35

Class 12 Sociology — Change and Development in Rural Society (Practice Quiz)

10 Qs · ~10 min

#36

Class 12 Sociology — The Constitution and Social Change (Practice Quiz)

10 Qs · ~10 min

#37

Class 12 Sociology — Cultural Change (Practice Quiz)

10 Qs · ~10 min

#38

Class 12 Sociology — Structural Change (Practice Quiz)

10 Qs · ~10 min

#39

Class 12 Sociology — The Challenges of Cultural Diversity (Practice Quiz)

10 Qs · ~10 min

#40

Class 12 Sociology — Patterns of Social Inequality and Exclusion (Practice Quiz)

10 Qs · ~10 min

#41

Class 12 Sociology — The Market as a Social Institution (Practice Quiz)

10 Qs · ~10 min

#42

Class 12 Sociology — Social Institutions: Continuity and Change (Practice Quiz)

10 Qs · ~10 min

#43

Class 12 Sociology — The Demographic Structure of the Indian Society (Practice Quiz)

10 Qs · ~10 min

#44

Class 12 Sociology — Introducing Indian Society (Practice Quiz)

10 Qs · ~10 min

#45

Class 12 Political Science — Recent Developments in Indian Politics (Practice Quiz)

10 Qs · ~10 min

#46

Class 12 Political Science — Regional Aspirations (Practice Quiz)

10 Qs · ~10 min

#47

Class 12 Political Science — The Crisis of Democratic Order (Practice Quiz)

10 Qs · ~10 min

#48

Class 12 Political Science — Challenges to and Restoration of the Congress System (Practice Quiz)

10 Qs · ~10 min

#49

Class 12 Political Science — India's External Relations (Practice Quiz)

10 Qs · ~10 min

#50

Class 12 Political Science — Politics of Planned Development (Practice Quiz)

10 Qs · ~10 min

#51

Class 12 Political Science — Era of One-party Dominance (Practice Quiz)

10 Qs · ~10 min

#52

Class 12 Political Science — Challenges of Nation Building (Practice Quiz)

10 Qs · ~10 min

#53

Class 12 Political Science — Environment and Natural Resources (Practice Quiz)

10 Qs · ~10 min

#54

Class 12 Political Science — Security in the Contemporary World (Practice Quiz)

10 Qs · ~10 min

#55

Class 12 Political Science — International Organisations (Practice Quiz)

10 Qs · ~10 min

#56

Class 12 Political Science — Contemporary South Asia (Practice Quiz)

10 Qs · ~10 min

#57

Class 12 Political Science — Contemporary Centres of Power (Practice Quiz)

10 Qs · ~10 min

#58

Class 12 Political Science — The End of Bipolarity (Practice Quiz)

10 Qs · ~10 min

#59

Class 12 Geography — Geographical Perspective on Selected Issues and Problems (Practice Quiz)

10 Qs · ~10 min

#60

Class 12 Geography — Planning and Sustainable Development in the Indian Context (Practice Quiz)

10 Qs · ~10 min

#61

Class 12 Geography — Mineral and Energy Resources (Practice Quiz)

10 Qs · ~10 min

#62

Class 12 Geography — Water Resources (Practice Quiz)

10 Qs · ~10 min

#63

Class 12 Geography — Land Resources and Agriculture (Practice Quiz)

10 Qs · ~10 min

#64

Class 12 Geography — Human Settlements (Practice Quiz)

10 Qs · ~10 min

#65

Class 12 Geography — Population: Distribution, Density, Growth and Composition (Practice Quiz)

10 Qs · ~10 min

#66

Class 12 Geography — International Trade (India) (Practice Quiz)

10 Qs · ~10 min

#67

Class 12 Geography — Transport and Communication (India) (Practice Quiz)

10 Qs · ~10 min

#68

Class 12 Geography — Tertiary and Quaternary Activities (Practice Quiz)

10 Qs · ~10 min

#69

Class 12 Geography — Secondary Activities (Practice Quiz)

10 Qs · ~10 min

#70

Class 12 Geography — Primary Activities (Practice Quiz)

10 Qs · ~10 min

#71

Class 12 Geography — Human Development (Practice Quiz)

10 Qs · ~10 min

#72

Class 12 Geography — The World Population: Distribution, Density and Growth (Practice Quiz)

10 Qs · ~10 min

#73

Class 12 Geography — Human Geography: Nature and Scope (Practice Quiz)

10 Qs · ~10 min

#74

Class 12 History — Framing the Constitution — The Beginning of a New Era (Practice Quiz)

10 Qs · ~10 min

#75

Class 12 History — Mahatma Gandhi and the Nationalist Movement — Civil Disobedience and Beyond (Practice Quiz)

10 Qs · ~10 min

#76

Class 12 History — Rebels and the Raj — The Revolt of 1857 and its Representations (Practice Quiz)

10 Qs · ~10 min

#77

Class 12 History — Colonialism and the Countryside — Exploring Official Archives (Practice Quiz)

10 Qs · ~10 min

#78

Class 12 History — Peasants, Zamindars and the State — Agrarian Society and the Mughal Empire (c. sixteenth-seventeenth centuries) (Practice Quiz)

10 Qs · ~10 min

#79

Class 12 History — An Imperial Capital: Vijayanagara (c. fourteenth to sixteenth century) (Practice Quiz)

10 Qs · ~10 min

#80

Class 12 History — Bhakti-Sufi Traditions — Changes in Religious Beliefs and Devotional Texts (c. eighth to eighteenth century) (Practice Quiz)

10 Qs · ~10 min

#81

Class 12 History — Through the Eyes of Travellers — Perceptions of Society (c. tenth to seventeenth century) (Practice Quiz)

10 Qs · ~10 min

#82

Class 12 History — Thinkers, Beliefs and Buildings — Cultural Developments (c. 600 BCE-600 CE) (Practice Quiz)

10 Qs · ~10 min

#83

Class 12 History — Kinship, Caste and Class — Early Societies (c. 600 BCE-600 CE) (Practice Quiz)

10 Qs · ~10 min

#84

Class 12 History — Kings, Farmers and Towns — Early States and Economies (c. 600 BCE-600 CE) (Practice Quiz)

10 Qs · ~10 min

#85

Class 12 History — Bricks, Beads and Bones — The Harappan Civilisation (Practice Quiz)

10 Qs · ~10 min

#86

Class 12 Economics — Open Economy Macroeconomics (Practice Quiz)

10 Qs · ~10 min

#87

Class 12 Economics — Government Budget and the Economy (Practice Quiz)

10 Qs · ~10 min

#88

Class 12 Economics — Determination of Income and Employment (Practice Quiz)

10 Qs · ~10 min

#89

Class 12 Economics — Money and Banking (Practice Quiz)

10 Qs · ~10 min

#90

Class 12 Economics — National Income Accounting (Practice Quiz)

10 Qs · ~10 min

#91

Class 12 Economics — Market Equilibrium (Practice Quiz)

10 Qs · ~10 min

#92

Class 12 Economics — The Theory of the Firm under Perfect Competition (Practice Quiz)

10 Qs · ~10 min

#93

Class 12 Economics — Production and Costs (Practice Quiz)

10 Qs · ~10 min

#94

Class 12 Economics — Theory of Consumer Behaviour (Practice Quiz)

10 Qs · ~10 min

#95

Class 12 Economics — Introduction (Practice Quiz)

10 Qs · ~10 min

#96

Class 12 Business Studies — Consumer Protection (Practice Quiz)

10 Qs · ~10 min

#97

Class 12 Business Studies — Marketing (Practice Quiz)

10 Qs · ~10 min

#98

Class 12 Business Studies — Financial Management (Practice Quiz)

10 Qs · ~10 min

#99

Class 12 Business Studies — Controlling (Practice Quiz)

10 Qs · ~10 min

#100

Class 12 Business Studies — Directing (Practice Quiz)

10 Qs · ~10 min

#101

Class 12 Business Studies — Staffing (Practice Quiz)

10 Qs · ~10 min

#102

Class 12 Business Studies — Organising (Practice Quiz)

10 Qs · ~10 min

#103

Class 12 Business Studies — Planning (Practice Quiz)

10 Qs · ~10 min

#104

Class 12 Business Studies — Business Environment (Practice Quiz)

10 Qs · ~10 min

#105

Class 12 Business Studies — Nature and Significance of Management (Practice Quiz)

10 Qs · ~10 min

#106

Class 12 Accountancy — Cash Flow Statement (Practice Quiz)

10 Qs · ~10 min

#107

Class 12 Accountancy — Accounting Ratios (Practice Quiz)

10 Qs · ~10 min

#108

Class 12 Accountancy — Analysis of Financial Statements (Practice Quiz)

10 Qs · ~10 min

#109

Class 12 Accountancy — Financial Statements of a Company (Practice Quiz)

10 Qs · ~10 min

#110

Class 12 Accountancy — Issue and Redemption of Debentures (Practice Quiz)

10 Qs · ~10 min

#111

Class 12 Accountancy — Accounting for Share Capital (Practice Quiz)

10 Qs · ~10 min

#112

Class 12 Accountancy — Dissolution of Partnership Firm (Practice Quiz)

10 Qs · ~10 min

#113

Class 12 Accountancy — Reconstitution of a Partnership Firm – Retirement/Death of a Partner (Practice Quiz)

10 Qs · ~10 min

#114

Class 12 Accountancy — Reconstitution of a Partnership Firm – Admission of a Partner (Practice Quiz)

10 Qs · ~10 min

#115

Class 12 Accountancy — Accounting for Partnership: Basic Concepts (Practice Quiz)

10 Qs · ~10 min

#116

Class 12 Maths — Probability (Practice Quiz)

10 Qs · ~10 min

#117

Class 12 Maths — Linear Programming (Practice Quiz)

10 Qs · ~10 min

#118

Class 12 Maths — Three Dimensional Geometry (Practice Quiz)

10 Qs · ~10 min

#119

Class 12 Maths — Vector Algebra (Practice Quiz)

10 Qs · ~10 min

#120

Class 12 Maths — Differential Equations (Practice Quiz)

10 Qs · ~10 min

#121

Class 12 Maths — Application of Integrals (Practice Quiz)

10 Qs · ~10 min

#122

Class 12 Maths — Integrals (Practice Quiz)

10 Qs · ~10 min

#123

Class 12 Maths — Application of Derivatives (Practice Quiz)

10 Qs · ~10 min

#124

Class 12 Maths — Continuity and Differentiability (Practice Quiz)

10 Qs · ~10 min

#125

Class 12 Maths — Determinants (Practice Quiz)

10 Qs · ~10 min

#126

Class 12 Maths — Matrices (Practice Quiz)

10 Qs · ~10 min

#127

Class 12 Maths — Inverse Trigonometric Functions (Practice Quiz)

10 Qs · ~10 min

#128

Class 12 Maths — Relations and Functions (Practice Quiz)

10 Qs · ~10 min

#129

Class 12 Biology — Biodiversity and its Conservation (Practice Quiz)

10 Qs · ~10 min

#130

Class 12 Biology — Ecosystem (Practice Quiz)

10 Qs · ~10 min

#131

Class 12 Biology — Organisms and Populations (Practice Quiz)

10 Qs · ~10 min

#132

Class 12 Biology — Biotechnology and its Applications (Practice Quiz)

10 Qs · ~10 min

#133

Class 12 Biology — Biotechnology: Principles and Processes (Practice Quiz)

10 Qs · ~10 min

#134

Class 12 Biology — Microbes in Human Welfare (Practice Quiz)

10 Qs · ~10 min

#135

Class 12 Biology — Human Health and Disease (Practice Quiz)

10 Qs · ~10 min

#136

Class 12 Biology — Evolution (Practice Quiz)

10 Qs · ~10 min

#137

Class 12 Biology — Molecular Basis of Inheritance (Practice Quiz)

10 Qs · ~10 min

#138

Class 12 Biology — Principles of Inheritance and Variation (Practice Quiz)

10 Qs · ~10 min

#139

Class 12 Biology — Reproductive Health (Practice Quiz)

10 Qs · ~10 min

#140

Class 12 Biology — Human Reproduction (Practice Quiz)

10 Qs · ~10 min

#141

Class 12 Biology — Sexual Reproduction in Flowering Plants (Practice Quiz)

10 Qs · ~10 min

#142

Class 12 Chemistry — Biomolecules (Practice Quiz)

10 Qs · ~10 min

#143

Class 12 Chemistry — Amines (Practice Quiz)

10 Qs · ~10 min

#144

Class 12 Chemistry — Aldehydes, Ketones and Carboxylic Acids (Practice Quiz)

10 Qs · ~10 min

#145

Class 12 Chemistry — Alcohols, Phenols and Ethers (Practice Quiz)

10 Qs · ~10 min

#146

Class 12 Chemistry — Haloalkanes and Haloarenes (Practice Quiz)

10 Qs · ~10 min

#147

Class 12 Chemistry — Coordination Compounds (Practice Quiz)

10 Qs · ~10 min

#148

Class 12 Chemistry — The d- and f-Block Elements (Practice Quiz)

10 Qs · ~10 min

#149

Class 12 Chemistry — Chemical Kinetics (Practice Quiz)

10 Qs · ~10 min

#150

Class 12 Chemistry — Electrochemistry (Practice Quiz)

10 Qs · ~10 min

#151

Class 12 Chemistry — Solutions (Practice Quiz)

10 Qs · ~10 min

#152

Class 12 Physics — Semiconductor Electronics: Materials, Devices and Simple Circuits (Practice Quiz)

10 Qs · ~10 min

#153

Class 12 Physics — Nuclei (Practice Quiz)

10 Qs · ~10 min

#154

Class 12 Physics — Atoms (Practice Quiz)

10 Qs · ~10 min

#155

Class 12 Physics — Dual Nature of Radiation and Matter (Practice Quiz)

10 Qs · ~10 min

#156

Class 12 Physics — Wave Optics (Practice Quiz)

10 Qs · ~10 min

#157

Class 12 Physics — Ray Optics and Optical Instruments (Practice Quiz)

10 Qs · ~10 min

#158

Class 12 Physics — Electromagnetic Waves (Practice Quiz)

10 Qs · ~10 min

#159

Class 12 Physics — Alternating Current (Practice Quiz)

10 Qs · ~10 min

#160

Class 12 Physics — Electromagnetic Induction (Practice Quiz)

10 Qs · ~10 min

#161

Class 12 Physics — Magnetism and Matter (Practice Quiz)

10 Qs · ~10 min

#162

Class 12 Physics — Moving Charges and Magnetism (Practice Quiz)

10 Qs · ~10 min

#163

Class 12 Physics — Electrostatic Potential and Capacitance (Practice Quiz)

10 Qs · ~10 min

#164

Class 12 Physics — Electric Charges and Fields (Practice Quiz)

10 Qs · ~10 min

#165

Class 12 Business Studies — Principles of Management (Practice Quiz)

10 Qs · ~10 min

#166

CBSE Class 12 — Genetics and Evolution (Practice Quiz)

10 Qs · ~10 min

#167

CBSE Class 12 — Matrices and Determinants (Practice Quiz)

10 Qs · ~10 min

#168

CBSE Class 12 — Solutions and Colligative Properties (Practice Quiz)

10 Qs · ~10 min

#169

Class 12 Physics — Current Electricity (Practice Quiz)

10 Qs · ~10 min

#170

CBSE Class 12 — Electrostatics and Electric Field (Practice Quiz)

10 Qs · ~10 min

#171

Humanities Subjects Practice Quiz | CBSE Class 12 Board Examination

10 Qs · ~10 min

Group Discussions

No forum posts available.

Easily Share with Your Tribe

Complete Summary and Solutions for Relations and Functions – NCERT Class XII Mathematics Part I, Chapter 1 – Types of Relations, Functions, Domain, Range, Types of Functions, Composition, Inverse Functions

Relations and Functions

Full Chapter Summary & Detailed Notes - Relations and Functions Class 12 NCERT

Overview & Key Concepts

1.1 Introduction

Conceptual Diagram: Relation vs Function

Why This Guide Stands Out

1.2 Types of Relations

Quick Table: Types of Relations

1.3 Types of Functions

1.4 Composition & Invertible

Miscellaneous Examples

Summary & Exercises

Key Definitions & Terms - Complete Glossary

Relation

Empty Relation

Universal Relation

Reflexive

Symmetric

Transitive

Equivalence Relation

Equivalence Class

One-One (Injective)

Onto (Surjective)

Bijective

Composition

Invertible Function

Domain/Codomain/Range

Solved Examples from NCERT - Step-by-Step with MathJax

Example 1: Empty/Universal in Boys School

Example 2: Congruent Triangles Equivalence

Example 3: Perpendicular Lines (Fig 1.1)

Example 4: Partial in {1,2,3}

Example 5: Even Diff in Z

Example 6: Odd/Even in {1-7}

Example 7: Roll Numbers One-One Not Onto

Example 9: \( f(x)=2x \) R→R Bijective

Example 11: \( x^2 \) Neither

Example 12: Piecewise Bijective

Example 13: Finite Onto ⇒ One-One

Example 15: Composition gof

Example 16: Non-Commute

Example 17: Invertible \( f(x)=4x+3 \)

Example 18: Intersection Equivalence

Example 19: Ordered Pairs xv=yu

Example 20: Mod 3 = Subsets

Example 21: Kernel Equivalence

Example 22: One-One Functions {1,2,3}→Self

Example 23: Relations Ref/Trans Not Sym

Example 24: Equiv Containing (1,2)(2,1)

Example 25: IN Onto, 2IN Not

Example 26: sin+cos Not One-One

Exercise 1.1 Questions & Answers - With MathJax Solutions

1(i): R in A={1..14}, 3x-y=0

1(ii): N, y=x+5, x<4

2: R={(a,b): a ≤ b²} Neither

3: {1-6}, b=a+1

4: ≤ Reflexive Trans Not Sym

5: ≤ b³

6: {(1,2),(2,1)} Sym Not Ref/Trans

7: Books Same Pages Equiv

8: |a-b| Even in {1-5} Equiv

9(i): |a-b| Multiple 4 in {0-12}

9(ii): a=b Identity Equiv

10: Examples

11: Distance from Origin Equiv

12: Similar Triangles Equiv

13: Same Sides Polygons Equiv

14: Parallel Lines Equiv

15: {1,2,3} R={(1,2),(2,3),(1,3),(3,1),(2,1),(3,2),(1,1),(2,2),(3,3)} Equiv

16: a=b-2, b>6 None

Exercise 1.2 Questions & Answers - With MathJax

1: \( f(x)=1/x \) R*→R* Bijection

2(i): N→N x² One-One Not Onto

2(ii): Z→Z x² Not One-One Onto? No

2(iii): R→R x² Not One-One Not Onto

2(iv): N→N x³ One-One Not Onto

2(v): Z→Z x³ Bijection

3: [x] Greatest Integer Neither

4: |x| Modulus Neither

1: \( f(x)=1/x \) R→R Bijection