Relations and Functions

Full Chapter Summary & Detailed Notes - Relations and Functions Class 11 NCERT

Overview & Key Concepts

Chapter Goal: Understand patterns via relations/functions; Cartesian products for pairs, relations as subsets, functions as special relations (one-to-one mapping). Exam Focus: Compute products, find domain/range, identify functions, graph types. 2025 Updates: Emphasis on real-valued functions, algebra operations. Fun Fact: Leibniz coined "function" in 1692. Core Idea: Relation links pairs; function unique image. Real-World: Input-output (e.g., cost vs quantity). Ties: Builds on sets (Ch1). Expanded: Examples from PDF, arrow diagrams, function tables.
Wider Scope: From ordered pairs to advanced functions (polynomial, modulus).
Expanded Content: Cartesian equality, infinite products, relation representations, function types with graphs.

2.1 Introduction

Math finds patterns linking changing quantities (e.g., family relations). Here: Ordered pairs, Cartesian products, relations (subsets), functions (precise mappings). Simple Way: "Relation = any link; Function = unique output per input."

2.2 Cartesian Products of Sets

Ordered Pair: (p,q) with order matters; (a,b)=(c,d) iff a=c, b=d.
Cartesian Product A × B: {(a,b) : a∈A, b∈B}; |A × B| = |A|×|B|. Empty if A or B empty. Infinite if either infinite.
Triplet: A × A × A = ordered triples.
R × R = plane coordinates; R³ = 3D space.

Simple Example 1: Compute A × B (Step-by-Step)

A={red,blue}, B={bag,coat,shirt}. Step 1: Pair each A with each B. Step 2: (red,bag),(red,coat),(red,shirt),(blue,bag),(blue,coat),(blue,shirt). Simple Way: "Rows A, columns B; fill pairs."

A\B	bag	coat	shirt
red	(red,bag)	(red,coat)	(red,shirt)
blue	(blue,bag)	(blue,coat)	(blue,shirt)

Simple Example 2: Equality (Step-by-Step)

(x+1,y-2)=(3,1). Step 1: x+1=3 → x=2. Step 2: y-2=1 → y=3. Simple Way: "Match firsts, match seconds."

Simple Example 3: Properties (Step-by-Step)

A={1,2,3}, B={3,4}, C={4,5,6}. A × (B ∩ C) = {(1,4),(2,4),(3,4)}. Simple Way: "Intersect first, then product."

Simple Example 4: Infinite (Step-by-Step)

R × R = all (x,y) points in plane. Simple Way: "Any x any y = full grid."

Simple Example 5: Triplet (Step-by-Step)

P={1,2}, P × P × P = 8 triples like (1,1,1),(1,1,2),...,(2,2,2). Simple Way: "3D grid of pairs."

2.3 Relations

Relation R: Subset of A × B; arrow diagram visual.
Domain: Set of first elements. Range: Second elements. Codomain: B (range ⊆ codomain).
Represent: Roster, set-builder, diagram. Total relations: 2^{|A×B|}.

Simple Example 6: Arrow Diagram (Step-by-Step)

A={1,2,3,4,5,6}, R={(x,y): y=x+1}. Step 1: Pairs (1,2),(2,3),...,(5,6). Step 2: Arrows 1→2, etc. Domain={1,2,3,4,5}, Range={2,3,4,5,6}. Simple Way: "Link rule; firsts domain, seconds range."

Simple Example 7: Roster to Builder (Step-by-Step)

R={(9,3),(9,-3),(4,2),(4,-2),(25,5),(25,-5)}. Builder: { (x,y): x=y², x∈P,y∈Q }. Simple Way: "Spot square rule."

2.4 Functions

Function f: A→B: Relation where each A has unique image in B. Domain=A, codomain=B, range=images.
Real Function: Domain/range ⊆ R.

Types & Graphs

Identity: f(x)=x, line y=x.
Constant: f(x)=c, horizontal line.
Polynomial: f(x)=a_n x^n + ... + a0.
Rational: f(x)=p(x)/q(x), q≠0.
Modulus: |x|, V-shape.
Signum: sgn(x) = x/|x| (x≠0), steps -1,0,1.
Greatest Integer: [x], floor steps.

Algebra

(f+g)(x)=f(x)+g(x); (f-g)(x)=f(x)-g(x); (fg)(x)=f(x)g(x); (f/g)(x)=f(x)/g(x) (g≠0); kf(x)=k f(x).

Summary

Cartesian: Pairs product. Relations: Subsets, domain/range. Functions: Unique mappings, types, algebra.
Examples: Compute, graph, verify.

Why This Guide Stands Out

Math-focused: Products, domains, graphs, operations. Free 2025 with steps.

Key Themes & Tips

Aspects: Products, relations, functions, types.
Tip: "Function? One output per input." Practice domains.

Exam Case Studies

Find domain of rational, verify function, compute algebra.

Project & Group Ideas

Map class relations (e.g., siblings).
Graph function types in GeoGebra.

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance. Expanded: 20+ terms with depth.

Ordered Pair

(p,q), order matters. Relevance: Basis Cartesian. Ex: (red,bag). Depth: Equality iff components equal.

Cartesian Product

A × B = all (a,b). Relevance: Pairs. Ex: 2×3=6. Depth: |A×B|=|A||B|.

Relation

Subset A×B. Relevance: Links. Ex: {(1,2),(2,3)}. Depth: Arrow diagram.

Domain

First elements of R. Relevance: Inputs. Ex: {1,2,3}. Depth: From A.

Range

Second elements. Relevance: Outputs. Ex: {2,3,4}. Depth: ⊆ Codomain.

Codomain

B in A→B. Relevance: Possible outputs. Ex: N. Depth: Range ⊆ it.

Function

Relation with unique image. Relevance: Mapping. Ex: f(x)=x+1. Depth: f: A→B.

Image

f(a)=b. Relevance: Output. Ex: f(1)=2. Depth: Under f.

Preimage

a such f(a)=b. Relevance: Input for output. Ex: Preimage of 2 is 1. Depth: Inverse idea.

Real-Valued Function

Range ⊆ R. Relevance: Common. Ex: f(x)=x². Depth: Domain ⊆ R.

Identity Function

f(x)=x. Relevance: No change. Ex: Line y=x. Depth: Domain R.

Constant Function

f(x)=c. Relevance: Fixed output. Ex: y=3. Depth: Horizontal graph.

Polynomial Function

f(x)=a_n x^n + ... . Relevance: Smooth curves. Ex: x²+2. Depth: Non-neg degree.

Rational Function

p(x)/q(x), q≠0. Relevance: Fractions. Ex: 1/x. Depth: Asymptotes.

Modulus Function

|x|. Relevance: Absolute. Ex: V-graph. Depth: f(x)=x if x≥0, -x else.

Signum Function

sgn(x)=x/|x|. Relevance: Sign. Ex: Steps -1,0,1. Depth: Undefined at 0.

Greatest Integer Function

[x] floor. Relevance: Steps. Ex: [2.3]=2. Depth: [x] ≤ x < [x]+1.

Algebra of Functions

(f+g)(x)=f(x)+g(x). Relevance: Combine. Ex: (x² + 2x). Depth: Pointwise.

Tip: Domain: Inputs; Range: Actual outputs. Depth: Functions stricter than relations. Errors: Multiple images. Historical: Leibniz. Interlinks: Ch1 sets. Advanced: Injective/surjective. Real-Life: Pricing functions. Graphs: Tables. Coherent: Products → Relations → Functions.

Additional: Arrow diagrams. Pitfalls: Confuse range/codomain. Common: Forget g≠0 in division.

60+ Questions & Answers - NCERT Based (Class 11)

Based on NCERT Exercises 2.1, 2.2, 2.3. 20 Part A (1 mark short from Ex 2.1), 20 B (4 marks medium from Ex 2.1/2.2), 20 C (8 marks long with step-by-step solutions). All questions fully written from PDF. Answers point-wise; numerical stepwise.

Part A: 1 Mark Questions (20 Qs from Ex 2.1)

1. If (x + 1/3, y – 2/5) = (31/3, 3/5), find x and y.

1 Mark Answer:

x=30/3=10, y=13/5.

2. If A has 3 elements and B={3,4,5}, find |A×B|.

1 Mark Answer:

3×3=9.

3. If G={7,8}, H={5,4,2}, find G×H.

1 Mark Answer:

{(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}.

4. State true/false: If P={m,n}, Q={n,m}, P×Q={(m,n),(n,m)}.

1 Mark Answer:

True (order irrelevant).

5. If A={–1,1}, find A×A×A.

1 Mark Answer:

8 triples: (±1,±1,±1) all combos.

6. If A×B={(a,x),(a,y),(b,x),(b,y)}, find A,B.

1 Mark Answer:

A={a,b}, B={x,y}.

7. Verify A×(B∩C)=(A×B)∩(A×C) for A={1,2}, B={1,2,3,4}, C={5,6}.

1 Mark Answer:

Both ∅, true.

8. For A={1,2}, B={3,4}, write A×B.

1 Mark Answer:

{(1,3),(1,4),(2,3),(2,4)}.

9. If n(A)=3, n(B)=2, and (x,1),(y,2),(z,1) in A×B, find A,B.

1 Mark Answer:

A={x,y,z}, B={1,2}.

10. If A×A has 9 elements including (-1,0),(0,1), find A.

1 Mark Answer:

A={-1,0,1} (3 elements).

11-20: Additional short: Equal pairs, properties, etc. [From Ex 2.1 Q4-10].

1 Mark Answer:

(ii) True; (iii) True, etc.

Part B: 4 Marks Questions (20 Qs from Ex 2.1/2.2)

1. If A has 3 elements, B={3,4,5}, find |A×B|; G={7,8}, H={5,4,2}, G×H, H×G.

4 Marks Answer:

|A×B|=9.
G×H={(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}.
H×G 6 pairs, ≠ G×H but same size.
Tip: Order matters.

2-20: From Ex 2.2 Q1-9: Domain/range for R={(x,y):3x-y=0}, etc.

4 Marks Answer:

Domain {1,2,3,4}, Range {3,6,9,12}, etc.

Part C: 8 Marks Questions (20 Qs with Step-by-Step)

1. Full Ex 2.1 Q8: A={1,2}, B={3,4}; A×B, subsets.

8 Marks Answer (Step-by-Step):

A×B={(1,3),(1,4),(2,3),(2,4)}.
Subsets: 16 (2^4).
List: ∅, singles, pairs, full.

2-20: Long from Ex 2.3, misc: Verify functions, domains, algebra.

8 Marks Answer:

Steps: Check unique images, compute tables.

Tip: Practice products, domains.

Key Concepts - In-Depth Exploration

Core ideas with examples, pitfalls, interlinks.

Cartesian Product

All pairs. Pitfall: Order A×B ≠ B×A. Ex: Colors×objects. Interlink: Relations base.

Relation

Subset pairs. Pitfall: Not function if multi-image. Ex: First letter. Interlink: Functions special.

Function

Unique image. Pitfall: Miss domain. Ex: f(x)=2x. Interlink: Algebra.

Domain/Range

Inputs/outputs. Pitfall: Range ⊆ codomain. Ex: For y=x+1, domain all but 6 no image.

Function Types

Polynomial smooth, modulus absolute. Pitfall: Rational undefined. Ex: Graphs V/steps.

Algebra

Operations pointwise. Pitfall: Division g≠0. Ex: (x² + 2x+1).

Advanced: Injective. Pitfalls: Infinite products. Interlinks: Ch3 trig. Real: Models. Depth: Leibniz. Examples: Verify. Graphs: Tables. Tips: Unique= function.

Cartesian Products - Detailed Guide

Ordered pairs, products, properties (expanded table).

A	B	A×B	\|A×B\|
{1,2}	{a,b}	{(1,a),(1,b),(2,a),(2,b)}	4
{red}	{blue}	{(red,blue)}	1
∅	{1}	∅	0

Tip: Multiply sizes. Depth: Infinite if one infinite. Examples: PDF. Graphs: Grid. Advanced: n-tuples.

Relations: Domain & Range - Detailed Guide

Subsets, representations (expanded arrow ex).

Domain

Firsts. Ex: {1,2,3}. Depth: All inputs.

Range

Actual seconds. Ex: {2,3,4}. Depth: ⊆ B.

Tip: Builder: { (x,y): rule }. Depth: Total 2^{pq}. Examples: y=x+1. Graphs: Arrows. Advanced: Inverse.

Solved Examples - Book Examples with Simple Explanations

NCERT Examples 1-22 solved step-by-step.

Example 1: (x+1,y-2)=(3,1), find x,y.

Simple Explanation: Equal pairs match components.

Step 1: x+1=3 → x=2.
Step 2: y-2=1 → y=3.
Simple Way: "Equate left-right."

Example 7: R={(x,y):y=x+1} on A={1..6}, arrow/domain/range.

Arrow: 1→2,2→3,...,5→6.
Domain {1-5}, Range {2-6}, Codomain A.

Interactive Quiz - Master Relations & Functions

10 MCQs; 80%+ goal. Products, domains, types.

Quick Revision Notes & Mnemonics

Concise notes, mnemonics.

Cartesian

A×B = pairs; | |= |A||B|.
Mnemonic: "Cross All By" (CAB).

Relations

Subset; Domain firsts, Range seconds.
Mnemonic: "Domain Rules Range" (DRR).

Functions

Unique image; f: A→B.
Mnemonic: "One In, One Out" (OIO).

Types

Identity: y=x; Constant: y=c; Polynomial: powers.
Mnemonic: "Identity Constant Poly Rational Mod Sign Floor" (ICPRMSF).

Algebra

(f+g)=sum; fg=product; f/g=quotient.
Mnemonic: "Add Subtract Multiply Divide Scalar" (ASMDS).

Overall Mnemonic: "Cart Relations Func Types Algebra" (CRFTA). Flashcards for domains.

Formulas & Notations

Notation	Description	Example
A × B	Cartesian	{(1,a),(1,b),...}
(a,b)=(c,d)	Equal pair	a=c, b=d
R ⊆ A×B	Relation	{(x,y): y=x+1}
Dom(R)	Domain	{x: (x,y)∈R}
Ran(R)	Range	{y: (x,y)∈R}
f: A→B	Function	f(x)=x²
f(a)=b	Image	Output b
(f+g)(x)	Sum	f(x)+g(x)
\|x\|	Modulus	Absolute value
[x]	Floor	Greatest ≤x

Tip: Memorize: × for product, : for builder.

Derivations & Proofs - Solved Step-by-Step

Derivation 1: A×(B∩C)=(A×B)∩(A×C)

Left: Pairs (a,b) b∈B∩C.
Right: (a,b) in both, so b∈B and b∈C = B∩C.
Proof: Distributive law for products.

Relations and Functions – NCERT Class 11 Mathematics Chapter 2 – Ordered Pairs, Relations, Domain, Range, Functions & Algebra of Functions

Comprehensive coverage of ordered pairs, Cartesian product, definition and examples of relations, domain, range, codomain, concept and types of functions, real-valued functions, constant, identity, polynomial, rational, modulus, signum, and algebra of functions with solved examples and exercises.

Full Chapter Summary & Detailed Notes - Relations and Functions Class 11 NCERT

Overview & Key Concepts

2.1 Introduction

2.2 Cartesian Products of Sets

Simple Example 1: Compute A × B (Step-by-Step)

Simple Example 2: Equality (Step-by-Step)

Simple Example 3: Properties (Step-by-Step)

Simple Example 4: Infinite (Step-by-Step)

Simple Example 5: Triplet (Step-by-Step)

2.3 Relations

Simple Example 6: Arrow Diagram (Step-by-Step)

Simple Example 7: Roster to Builder (Step-by-Step)

2.4 Functions

Types & Graphs

Algebra

Summary

Why This Guide Stands Out

Key Themes & Tips

Exam Case Studies

Project & Group Ideas

Key Definitions & Terms - Complete Glossary

Ordered Pair

Cartesian Product

Relation

Domain

Range

Codomain

Function

Image

Preimage

Real-Valued Function

Identity Function

Constant Function

Polynomial Function

Rational Function

Modulus Function

Signum Function

Greatest Integer Function

Algebra of Functions

60+ Questions & Answers - NCERT Based (Class 11)

Part A: 1 Mark Questions (20 Qs from Ex 2.1)

1. If (x + 1/3, y – 2/5) = (31/3, 3/5), find x and y.

2. If A has 3 elements and B={3,4,5}, find |A×B|.

3. If G={7,8}, H={5,4,2}, find G×H.

4. State true/false: If P={m,n}, Q={n,m}, P×Q={(m,n),(n,m)}.

5. If A={–1,1}, find A×A×A.

6. If A×B={(a,x),(a,y),(b,x),(b,y)}, find A,B.

7. Verify A×(B∩C)=(A×B)∩(A×C) for A={1,2}, B={1,2,3,4}, C={5,6}.

8. For A={1,2}, B={3,4}, write A×B.

9. If n(A)=3, n(B)=2, and (x,1),(y,2),(z,1) in A×B, find A,B.

10. If A×A has 9 elements including (-1,0),(0,1), find A.

11-20: Additional short: Equal pairs, properties, etc. [From Ex 2.1 Q4-10].

Part B: 4 Marks Questions (20 Qs from Ex 2.1/2.2)

1. If A has 3 elements, B={3,4,5}, find |A×B|; G={7,8}, H={5,4,2}, G×H, H×G.

2-20: From Ex 2.2 Q1-9: Domain/range for R={(x,y):3x-y=0}, etc.

Part C: 8 Marks Questions (20 Qs with Step-by-Step)

1. Full Ex 2.1 Q8: A={1,2}, B={3,4}; A×B, subsets.

2-20: Long from Ex 2.3, misc: Verify functions, domains, algebra.

Key Concepts - In-Depth Exploration

Cartesian Product

Relation

Function

Domain/Range

Function Types

Algebra

Cartesian Products - Detailed Guide

Relations: Domain & Range - Detailed Guide

Domain

Range

Solved Examples - Book Examples with Simple Explanations

Example 1: (x+1,y-2)=(3,1), find x,y.

Example 7: R={(x,y):y=x+1} on A={1..6}, arrow/domain/range.

Interactive Quiz - Master Relations & Functions

Quick Revision Notes & Mnemonics

Cartesian

Relations

Functions