Relations and Functions — Study Notes

1. Introduction

Relations and functions form the bedrock of modern mathematics. A relation describes how the elements of one set are connected to the elements of another set through a rule, whereas a function is a special kind of relation in which every element of the domain is paired with exactly one element of the codomain. This chapter revisits and deepens the Class 11 ideas of Cartesian products, relations, and functions by classifying relations as reflexive, symmetric, transitive, and equivalence relations, and by classifying functions as one-one, onto, and bijective.

The chapter also introduces the composition of functions and the inverse of a bijective function. These concepts are not merely abstract; they power the definition of inverse trigonometric functions, the change of variables in integration, and the solution of differential equations. A firm command of equivalence classes and binary operations additionally prepares students for the study of algebra and number theory at higher levels.

Mastering this chapter is essential because virtually every later chapter assumes fluency with functions. The notation $f: A \to B$, the ability to check injectivity and surjectivity, and the technique of composing functions reappear throughout calculus and coordinate geometry. Treat this chapter as the language toolkit for the entire book.

2. Types of Relations

A relation R in a set A is a subset of A x A. That is, $R \subseteq A \times A$. If $(a, b) \in R$, we write $a\,R\,b$ and say that a is related to b.

Reflexive Relation

A relation R in A is reflexive if every element is related to itself: $a\,R\,a$ for all $a \in A$. Equivalently, $(a, a) \in R$ for every $a \in A$.

Symmetric Relation

A relation R in A is symmetric if $a\,R\,b$ implies $b\,R\,a$. In other words, whenever $(a, b) \in R$, we must also have $(b, a) \in R$.

Transitive Relation

A relation R in A is transitive if $a\,R\,b$ and $b\,R\,c$ together imply $a\,R\,c$ for all $a, b, c \in A$.

Equivalence Relation

A relation R in A is an equivalence relation if it is simultaneously reflexive, symmetric, and transitive. The most familiar example is equality itself on the set of real numbers.

Example

Consider the relation R on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. This relation is reflexive because all diagonal pairs are present, symmetric because (1, 2) and (2, 1) both occur, and transitive because the only non-trivial chains lead to pairs already listed. Hence R is an equivalence relation.

3. Equivalence Classes

Let R be an equivalence relation in a set A. For any element $a \in A$, the equivalence class of a, denoted $[a]$, is the set of all elements of A that are related to a:

$$[a] = {x \in A : x\,R\,a}$$

Every element of A belongs to exactly one equivalence class. Two equivalence classes are either identical or disjoint. The equivalence classes partition the set A into mutually disjoint, non-empty subsets whose union is A.

4. Types of Functions

One-One (Injective) Functions

A function $f: A \to B$ is one-one if distinct elements of A have distinct images in B: $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Equivalently, $x_1 \neq x_2$ implies $f(x_1) \neq f(x_2)$.

Onto (Surjective) Functions

A function $f: A \to B$ is onto if every element of B has at least one pre-image in A, that is, the range of f equals the codomain B. For every $b \in B$ there exists $a \in A$ such that $f(a) = b$.

Bijective Functions

A function that is both one-one and onto is called bijective, or a one-to-one correspondence.

Counting Functions

If n(A) = p and n(B) = q, then: - The number of functions from A to B is $q^p$. - The number of one-one functions from A to B is $qP_p = \frac{q!}{(q-p)!}$ if $q \geq p$, otherwise zero. - The number of onto functions from A to B, when $q \leq p$, is $\sum_{k=0}^{q} (-1)^k \binom{q}{k} (q-k)^p$.

A function from a finite set A to itself is one-one if and only if it is onto.

5. Composition of Functions

Let $f: A \to B$ and $g: B \to C$ be two functions. The composition of g with f, written $g \circ f$, is the function from A to C defined by

$$(g \circ f)(x) = g(f(x)) \quad \text{for all } x \in A$$

Properties of Composition

• The composition of two functions is always a function.
• Composition is associative: $h \circ (g \circ f) = (h \circ g) \circ f$.
• Composition is generally not commutative; $f \circ g \neq g \circ f$ in most cases.
• If $f: A \to B$ and $g: B \to C$ are both one-one, then $g \circ f$ is one-one.
• If $f$ and $g$ are both onto, then $g \circ f$ is onto.
• If $g \circ f$ is one-one, then f is one-one (but g need not be).
• If $g \circ f$ is onto, then g is onto (but f need not be).

6. Inverse of a Function

A function $f: A \to B$ has an inverse if and only if f is bijective. The inverse function $f^{-1}: B \to A$ satisfies

$$f^{-1}(y) = x \iff f(x) = y$$

The inverse function satisfies the fundamental relations $f^{-1} \circ f = I_A$ and $f \circ f^{-1} = I_B$, where $I_A$ and $I_B$ are the identity functions on A and B respectively.

Steps to Find the Inverse

1. Write $y = f(x)$.
2. Solve this equation for x in terms of y.
3. Interchange x and y to obtain $y = f^{-1}(x)$.

For example, if $f(x) = \frac{2x + 3}{x - 2}$, then setting $y = \frac{2x + 3}{x - 2}$ gives $y(x - 2) = 2x + 3$, so $xy - 2y = 2x + 3$, hence $x(y - 2) = 2y + 3$, and $x = \frac{2y + 3}{y - 2}$. Interchanging variables, $f^{-1}(x) = \frac{2x + 3}{x - 2}$, which in this case equals f itself, a property called being an involution.

7. Binary Operations

A binary operation * on a set A is a function $*: A \times A \to A$ that assigns to each ordered pair $(a, b)$ of elements of A a unique element $a * b$ of A.

Properties of Binary Operations

• Commutative: $a * b = b * a$ for all $a, b \in A$.
• Associative: $(a * b) * c = a * (b * c)$ for all $a, b, c \in A$.
• Identity element: An element $e \in A$ such that $a * e = e * a = a$ for all $a \in A$.
• Inverse: An element $b \in A$ such that $a * b = b * a = e$, where e is the identity.

Number of Binary Operations

If A has n elements, then the number of binary operations on A is $n^{n^2}$, since each of the $n^2$ ordered pairs can be assigned any of the n elements.

Quick Revision Tables

Table 1: Properties of Relations

Table 2: Types of Functions

Table 3: Composition and Inverse Quick Facts

Mind Map

Important Diagrams (SVG)

Diagram 1: Injective, Surjective and Bijective Mappings

Diagram 2: Composition of Functions f : A to B and g : B to C

Common Mistakes

1. Declaring a relation reflexive while only some diagonal elements (a, a) are present. Reflexivity demands $(a, a) \in R$ for every single element of A.
2. Confusing symmetric with reflexive. Symmetric means a R b implies b R a; it does not require a R a.
3. Forgetting that to be transitive, the relation must close every chain; a single counterexample with $a\,R\,b$ and $b\,R\,c$ but not $a\,R\,c$ destroys transitivity.
4. Calling a function one-one when it is only onto, or vice versa. The two properties are independent except on finite sets mapping to themselves.
5. Using the number of onto functions formula incorrectly when the codomain is larger than the domain. If n(B) > n(A), no onto function exists.
6. Writing $f^{-1}(x) = \frac{1}{f(x)}$. The symbol $f^{-1}$ means the inverse function, not the reciprocal.
7. Computing $g \circ f$ as $f(g(x))$ instead of $g(f(x))$. Order matters: apply the function closest to x first.
8. Claiming the inverse exists when f is only one-one or only onto. Both injectivity and surjectivity are required.
9. Assuming all binary operations are commutative and associative. Subtraction on Z is neither commutative nor associative.
10. Forgetting that $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$, not $f^{-1} \circ g^{-1}$.

Exam Tips

1. To test reflexivity, write $(a, a)$ pairs for every element of the set; to test symmetry, swap every pair; to test transitivity, verify the implication on all chains.
2. In MCQs, test injectivity by assuming $f(x_1) = f(x_2)$ and showing $x_1 = x_2$; test surjectivity by solving $f(x) = y$ for arbitrary y.
3. For the number of relations in a set A, remember it is $2^{n(A)^2}$, not $2^{n(A)}$.
4. To find the inverse of a bijective function, interchange x and y and solve; then verify by checking $f(f^{-1}(x)) = x$.
5. For composition questions, first identify inner and outer functions; always evaluate the inner function completely before the outer one.
6. On a finite set A with f : A to A, if f is one-one then it is automatically onto; use this shortcut in objective questions.
7. When asked for the equivalence classes, pick one representative and collect everything related to it; verify the classes are disjoint.
8. Practise the standard involutions such as $f(x) = \frac{ax+b}{cx-a}$ and the identity $f \circ g = g \circ f$ cases to answer inverse questions quickly.

Conclusion

Relations and functions convert intuitive ideas of dependence into precise mathematical statements. The classification of relations as reflexive, symmetric, transitive, and equivalence relations, together with equivalence classes, gives structure to sets. Functions, classified as injective, surjective, and bijective, form the basis for composition and inversion, which are used in every branch of higher mathematics. The notion of a binary operation completes the toolkit by turning sets into algebraic structures. A student who has internalised the language of this chapter will find inverse trigonometric functions, change of variables in integration, and the concept of invertible matrices natural extensions of these same ideas. This chapter is therefore not just the first chapter of the book but the conceptual foundation on which the entire course is built.