Introduction

1.1 Introduction

In Class XI you met the fundamental ideas of relations and functions — domain, co-domain, and range — and studied real-valued functions such as polynomial, rational, trigonometric, exponential, and logarithmic functions along with their graphs. That foundation is the starting point for this chapter.

The word "relation" in everyday English refers to a link between two objects — "Amit is the brother of Bina." Mathematics borrows this idea but makes it precise. Let $A$ be the set of all students in Class XII of a school and $B$ the set of all students in Class XI of the same school. Between these two sets we can think of several natural relations:

• $\{(a, b) \in A \times B : a \text{ is brother of } b\}$
• $\{(a, b) \in A \times B : a \text{ is sister of } b\}$
• $\{(a, b) \in A \times B : \text{age of } a \text{ is greater than age of } b\}$
• $\{(a, b) \in A \times B : \text{total marks of } a \text{ is less than total marks of } b\}$
• $\{(a, b) \in A \times B : a \text{ lives in the same locality as } b\}$

Each is a collection of ordered pairs $(a, b)$, with $a \in A$ and $b \in B$, satisfying a specific condition. This leads to the formal definition.

A relation $R$ from a set $A$ to a set $B$ is a subset of the Cartesian product $A \times B$.

Thus $R \subseteq A \times B$. If $(a, b) \in R$, we say "$a$ is related to $b$ under $R$" and write $a\,R\,b$; if $(a, b) \notin R$, we write $a\,\cancel{R}\,b$. Crucially, the definition requires no "recognisable connection" between $a$ and $b$ — any subset of $A \times B$ qualifies as a relation, and this abstraction is what gives the concept its power.

As you know from Class XI, a function $f : A \to B$ is a special relation in which every element of $A$ is related to exactly one element of $B$.

In this chapter we go deeper, studying:

• Types of relations (reflexive, symmetric, transitive, and equivalence relations).
• Types of functions (injective, surjective, and bijective).
• Composition of functions and invertible functions.
• Binary operations.

The chapter opens with a portrait of Lejeune Dirichlet (1805–1859), who gave the modern definition of a function as a rule assigning a unique output to each input — freeing the concept from being tied to a single algebraic formula.

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