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

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.

Miscellaneous Exercise Questions & Answers

7 Qs + Summary; solutions.

1: \( f(x) = x + \frac{1}{|x|} \) One-One Onto to (-1,1)

Solution: Strict increase; range (-1,1).

2: x³ Injective

Solution: Strict mono.

3: ⊂ on P(X) Not Equiv

Solution: Ref? No A⊂A false? Wait, proper? No, reflexive yes but not sym/trans.

4: Onto to Self n! / (n-k)!k! Wait, Surjections

Solution: Stirling 2nd kind S(n,n) * n! but to self bijections n!.

5: f(x)=x²-x, g(x)=(2x-1)/2 Equal? No

Solution: f(0)=0, g(0)= -1/2 no.

6: Relations Ref Sym Not Trans Containing (1,2)(1,3)

Solution: (C) 3.

7: Equiv Containing (1,2)

Solution: (B) 2.

Tip: Advanced proofs.

Interactive Quiz - Master Relations & Functions

10 MCQs; 80%+ goal. Covers defs, examples.

Quick Revision Notes & Mnemonics

Concise summaries in tables. Bold keys; short phrases.

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".

Overall Tip: RST-IOB-CI-DU scan (3 mins). Flashcards: Term + Ex.