Chapter 2: Relations and Functions

If sets are the nouns of mathematics, then relations and functions are the verbs. They tell us how elements of one set are linked to elements of another. The Class XI treatment moves us from a primary-school idea of "mapping" to a precise definition that will support every calculus and probability statement to come.

The core idea is the ordered pair $(a, b)$. Unlike a set, the order matters: $(2, 3) \ne (3, 2)$. From ordered pairs we build the Cartesian product $A \times B$, every possible pair with first coordinate in $A$ and second in $B$. A relation $R$ from $A$ to $B$ is just a subset of $A \times B$ , a chosen list of which $a$ "relates to" which $b$.

A function is a special relation: one with the rule that every $a \in A$ is paired with exactly one $b \in B$. This precision matters. "Father of" is a function from people to people (each person has one father). "Sibling of" is a relation but not a function , a person may have many siblings, or none. The single-output rule is what makes calculation possible.

In Class XI we focus on real-valued functions of a real variable , i.e. $f : D \subseteq \mathbb{R} \to \mathbb{R}$. You will catalogue the graphs of the polynomial, modulus, signum, greatest-integer, square-root and reciprocal functions, and learn how to compute their domains and ranges. You will also see the algebra of functions: $f + g$, $f - g$, $fg$, $f/g$ , built pointwise.

For Class XII and JEE, this chapter is the language in which every subsequent topic is spoken: limits, derivatives, integrals, differential equations, and probability all start with "let $f : \dots$". The fluency you build here decides how quickly you make sense of the entire syllabus to come.

What's inside

1. Ordered pairs and Cartesian products , building blocks of relations.
2. Relations , domain, range, codomain and arrow diagrams.
3. Functions: definition and notation , when a relation is a function.
4. Real-valued functions and their graphs , polynomial, modulus, signum, greatest-integer, square root, reciprocal.
5. Domain and range of real functions , practical recipes.
6. Algebra of real functions , sum, difference, product, quotient.

Key results / Formula card

How to read this chapter

Begin with ordered pairs , they are the atom from which everything is built. Always draw an arrow diagram (or graph) for any new relation: you will see at once whether it is a function. When computing domains, list every restriction (denominators, square roots, logarithms) and take the intersection. By the end you should be able to read any real-function expression and instantly state its domain.