Chapter 1 · Introduction

Chapter 1 — Relations and Functions

Relations that behave like equality sort a set into classes; functions that are one-one and onto pair two sets off perfectly. Both ideas return all year.

Every rule that links things together is a relation: “is a sibling of”, “has the same PIN code as”, “divides”, “is parallel to”. In Class 11 you described such links as sets of ordered pairs. This chapter asks a sharper question — what kind of link is it? Some relations behave exactly like the equals sign: everything is related to itself, links work both ways, and links can be chained. Those relations quietly sort a whole set into tidy, non-overlapping groups, the way a calendar sorts all dates into seven weekdays or a clock sorts all hours into twelve.

Functions are special relations, and they too come in types. Does a function ever send two inputs to the same output? Does it reach every possible output? A function that does neither of the bad things — a bijection — can be run backwards. That single idea is the doorway to Chapter 2, where we restrict sine, cosine and tangent until they become bijections and can be inverted, and it sits behind every “solve for $x$” you will do in calculus.

Relations on a set→Reflexive · symmetric · transitive→Equivalence classes→One-one & onto→Bijections→Inverse trig (Ch 2)
🎯 By the end of this chapter you can…
  • recognise empty and universal relations and test any relation for the reflexive, symmetric and transitive properties, with a proof or a counterexample;
  • prove that a relation is an equivalence relation and list or describe its equivalence classes;
  • decide whether a function between finite sets, or on $\mathbb N$, $\mathbb Z$ or $\mathbb R$, is one-one, onto or bijective, and justify it algebraically and with the horizontal line test;
  • use counting (the pigeonhole principle) to decide what is possible between finite sets;
  • answer the Board's 1-, 3- and 5-mark questions on this chapter and the JEE-style MCQs that build on it.
Teaser — When Is a Relation a Function, and What Kind?

The grid is $A\times B$ with $A=\{1,2,3,4\}$ (rows) and $B=\{p,q,r,s\}$ (columns); a dot is a pair in the relation. Play draws random relations — or click cells yourself. Read the rows to decide function or not (exactly one dot in every row), then the columns to decide one-one (at most one dot per column) and onto (at least one per column).

a function A → B?—
one-one?—
onto?—

Sections in this chapter

1.1

Types of Relations

Empty and universal relations; reflexive, symmetric and transitive; equivalence relations and their classes.

$[a]=\{x:(x,a)\in R\}$
Open section →1.2

Types of Functions

One-one, onto and bijective functions on finite sets and on ℝ; the horizontal line test.

$f(x_1)=f(x_2)\Rightarrow x_1=x_2$
Open section →
✅ Before you start
You should be comfortable with sets, subsets and interval notation (Class 11 Mathematics, §1.1 and §1.2), Cartesian products and relations (Class 11, §2.1 and §2.2), and functions with their domain, codomain, range and graphs — including $|x|$, $[x]$ and the signum function (Class 11, §2.3). Quick self-test: if $A=\{1,2\}$, can you list all four pairs of $A\times A$ without looking?
← Welcome to Class 12 Mathematics1.1 Types of Relations →