Chapter 2 · Introduction

Chapter 2 — Relations and Functions

Pair things up, link them, and send every input to exactly one output — ordered pairs, Cartesian products, relations and functions, the language in which the rest of school mathematics describes change.

Every row of your school's timetable pairs a period with a subject; every entry in a phone's contact list pairs a name with a number; a railway reservation chart pairs each seat with a passenger. In each case the order matters — "(Period 3, Physics)" is not the same as "(Physics, Period 3)" — and in each case we are interested in some pairs and not others. The mathematics of such pairings begins with the ordered pair and the set of all possible pairs, the Cartesian product $A\times B$ — named after René Descartes, whose idea of labelling every point of a plane by a pair of numbers you used throughout Class 10 coordinate geometry.

A relation picks out the pairs we care about: "is a factor of", "is the capital of", "was born in the same month as". A function is a relation with one strict rule — every input has exactly one output — and that rule is what makes functions so useful: a vending machine, a tax calculator or the formula $s=ut+\tfrac12at^2$ would be useless if one input could give two answers. In this chapter you will meet the standard real functions — identity, constant, polynomial, rational, modulus, signum and greatest integer — draw their graphs, and learn to add, subtract, multiply and divide functions. Everything in Chapter 3 (trigonometric functions) and in Chapter 12 (limits and derivatives) is built on this foundation.

Pair: ordered pairs, $A\times B$→Select: relations as subsets of $A\times B$→Restrict: one output per input — functions→Picture and combine: graphs, $f\pm g$, $fg$, $f/g$

🎯 By the end of this chapter you can…

use equality of ordered pairs, write $A\times B$ in roster form, and find $n(A\times B)=n(A)\cdot n(B)$;
describe a relation from $A$ to $B$ in roster form, set-builder form and as an arrow diagram, and find its domain, codomain and range;
count the relations from $A$ to $B$, and decide whether a given relation is a function;
find the domain and range of a real function, and sketch the graphs of the identity, constant, polynomial, rational, modulus, signum and greatest integer functions;
form $f+g$, $f-g$, $fg$, $f/g$ and $\alpha f$, and state the domain of each.

On the left, each input $x$ sends an arrow to its output. On the right, the same arrow becomes an ordered pair $(x,y)$ — a point. After the listed inputs, the rule is applied to every real number in between and the points fuse into a curve: the graph. Try the last rule: one input sends out two arrows, so it is a relation but not a function — and its picture fails the "one point above each $x$" test.

Rule:

now—

ordered pairs so far—

function?—

Sections in this chapter

2.1

Cartesian Products of Sets

Ordered pairs and their equality, the product $A\times B$, counting its elements, and the plane $\mathbb{R}\times\mathbb{R}$.

$n(A\times B)=n(A)\cdot n(B)$

Open section →2.2

Relations

A relation as a subset of $A\times B$; arrow diagrams; domain, codomain and range; counting relations.

$R\subset A\times B$

Open section →2.3

Functions and Their Graphs

Functions as special relations, real functions, and the graphs of the standard functions from identity to $[x]$.

$f(x)=[x],\ \ f(x)=|x|$

Open section →2.4

Algebra of Real Functions

Sum, difference, product, quotient and scalar multiple of functions, with their domains.

$(f+g)(x)=f(x)+g(x)$

Open section →

✅ Before you start

You need the language of sets from Chapter 1: roster and set-builder forms (Section 1.1), subsets, intervals such as $[a,b)$ and $(-\infty,b]$ (Section 1.2), and union, intersection and difference (Section 1.3) — domains of functions are written with exactly these tools. From Class 10 Mathematics you need coordinate geometry (plotting points, Chapter 7), polynomials and their zeroes (Chapter 2), and linear and quadratic equations (Chapters 3 and 4), since finding where a denominator is zero is part of finding a domain.

← 1.3 Venn Diagrams and Operations on Sets 2.1 Cartesian Products of Sets →