Chapter 1: Relations and Functions

In Class XI you learned to think of a relation as a subset of $A \times B$ and a function as a special relation in which every input has exactly one output. In Class XII we sharpen those ideas into tools. We ask precise structural questions: when does a relation partition a set neatly into classes? When can a function be inverted? When are two functions equal? These are not pedantic distinctions , they are the language used everywhere from group theory to machine learning.

The first half of the chapter takes a relation $R$ on a set $A$ and tests it for three properties , reflexive, symmetric, transitive. A relation that satisfies all three is an equivalence relation; it slices the set into disjoint equivalence classes. The second half treats functions as objects in their own right. We classify them as injective (one-one), surjective (onto), or bijective (both). Only bijections admit inverses, and the chapter culminates with the formal definition of $f^{-1}$ and the rule $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

For board examinations the most common questions ask you to verify that a given relation is or is not an equivalence relation, or to prove that a particular function is bijective and find its inverse. For JEE the same ideas reappear inside problems on functional equations, periodicity, and counting (how many functions from a set of size $m$ to a set of size $n$ are injective?). A confident grasp of this chapter pays dividends throughout calculus, where every theorem about $f$ secretly assumes you know what kind of map $f$ is.

The prerequisites are light: set notation, Cartesian product, the definition of a function from Class XI, and basic algebra. What is new is the style of argument. You will write more proofs in this chapter than you did in all of Class XI, and you will learn the discipline of "let $x \in A$ be arbitrary; we show ...". Treat every definition as a checklist: to prove $R$ is symmetric, you must show that for all $a, b$, if $(a, b) \in R$ then $(b, a) \in R$. Skipping the universal quantifier is the single most common error.

A guiding philosophy: a relation is "good" when it preserves structure, and a function is "good" when it has an inverse. The chapter is really a tour of what "good" means in different settings.

What's inside

1. Types of relations , empty, universal, reflexive, symmetric, transitive.
2. Equivalence relations , equivalence classes and partitions.
3. Types of functions , one-one, onto, bijective.
4. Composition of functions , building new maps from old ones.
5. Invertible functions , when and how to invert a map.
6. Worked problems mixing the ideas , equivalence + functions together.
7. Pitfalls and JEE-style traps , counter-examples worth memorising.

Key results / Formula card

A bijection from a set of size $n$ to itself can be made in $n!$ ways. Injections from a set of size $m$ into a set of size $n$ (with $m \le n$) number $\dfrac{n!}{(n-m)!}$.

How to read this chapter

Work each subtopic with pen and paper. For every theorem, try to construct a counter-example that fails one hypothesis and see what breaks. Memorising definitions without testing them on examples is the surest way to lose marks.