Relations and Functions notes: ISC Class 12 Mathematics

1. Relations on a set

A relation on $A$ is a subset of $A \times A$ . If $n(A) = n$ , there are $n^2$ ordered pairs and $2^{n^2}$ relations.

Empty relation: $\varnothing$ . Universal relation: $A \times A$ . Identity relation: $I_A = \{(a, a) : a \in A\}$ .

The smallest, the identity and the largest relation on $A$.
Relation on $A = \{p, q\}$	Ordered pairs	How many
Empty	$\varnothing$	$0$
Identity $I_A$	$\{(p, p), (q, q)\}$	$2$
Universal $A \times A$	$\{(p, p), (p, q), (q, p), (q, q)\}$	$4$

Formulas

Counting relations
$\text{number of relations on } A = 2^{n^2}$

Key terms

Relation on A: A subset of $A \times A$ .
Identity relation: $\{(a, a) : a \in A\}$ .

2. Reflexive, symmetric, transitive and equivalence relations

Reflexive: $(a, a) \in R$ for every $a \in A$ .
Symmetric: $(a, b) \in R \Rightarrow (b, a) \in R$ .
Transitive: $(a, b) \in R,\ (b, c) \in R \Rightarrow (a, c) \in R$ .

All three together make an equivalence relation. Its classes $[a] = \{x \in A : x\,R\,a\}$ are disjoint and cover $A$ (a partition). "Same value of a property" (same remainder, same instrument) always gives an equivalence relation.

To prove a property, use general letters; to disprove one, give one concrete counterexample.

One counterexample is enough to break a property.
Property	What to check	One failure looks like
Reflexive	every $(a, a)$ is present	a missing loop
Symmetric	each pair has its reverse	$(a, b)$ without $(b, a)$
Transitive	each chain $(a, b), (b, c)$ has $(a, c)$	a chain without its shortcut

The classes do not overlap and together make up the whole set.
Remainder on division by $4$	Class
$1$	$\{1, 5, 9\}$
$2$	$\{2, 6, 10\}$
$3$	$\{3, 7\}$
$0$	$\{4, 8\}$

Formulas

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

Key terms

Equivalence relation: Reflexive, symmetric and transitive.
Equivalence class: $[a]$ : all elements related to $a$ .

3. Functions, domain and range

A function $f : X \to Y$ assigns to each $x \in X$ exactly one $f(x) \in Y$ . Domain $X$ , codomain $Y$ , range $\{f(x) : x \in X\} \subseteq Y$ .

For real functions: exclude values that make a denominator zero or put a negative number under an even root. Find the range by solving $y = f(x)$ for $x$ or by completing the square (for example $x^2 - 4x + 9 = (x - 2)^2 + 5 \ge 5$ ).

Domain: where the formula makes sense. Range: the values it actually takes.
$f(x)$	Domain	Range
$\sqrt{x - 3}$	$[3, \infty)$	$[0, \infty)$
$\dfrac{1}{x - 2}$	$\mathbb{R} - \{2\}$	$\mathbb{R} - \{0\}$
$x^2 + 1$	$\mathbb{R}$	$[1, \infty)$
$\|x\| - 2$	$\mathbb{R}$	$[-2, \infty)$

$y = \sqrt{16 - x^2}$

Domain $[-4, 4]$, range $[0, 4]$.

Key terms

Range: The set of values the function actually takes.

4. One-one, many-one, onto and into functions

One-one: $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$ . Otherwise many-one.
Onto: every $y \in Y$ has a pre-image (range $= Y$ ). Otherwise into.
Bijective: one-one and onto.

Finite sets with $n(X) = m$ , $n(Y) = n$ : $n^m$ functions; $n(n - 1)\cdots(n - m + 1)$ one-one functions when $m \le n$ ; onto functions to a $2$ -element set: $2^m - 2$ . A horizontal line meeting a graph twice shows "many-one".

Every function is one-one or many-one, and onto or into.
Type	Meaning	Arrow picture
one-one	different inputs, different outputs	no element of $Y$ gets two arrows
many-one	some output has two or more inputs	some element of $Y$ gets two arrows
onto	range $=$ codomain	every element of $Y$ gets an arrow
into	range $\ne$ codomain	some element of $Y$ gets no arrow

$y = x^2 - 6x$
$y = -5$

Two hits on one horizontal line: the function is many-one.

Formulas

One-one
$f(x_1) = f(x_2) \Rightarrow x_1 = x_2$

Key terms

Into function: A function whose range is smaller than its codomain.
Many-one function: Some output has two or more pre-images.

5. Composite functions

$(f \circ g)(x) = f(g(x))$ : apply $g$ first. It needs the range of $g$ to lie in the domain of $f$ (restrict $x$ where necessary).

Not commutative: usually $f \circ g \ne g \circ f$ .
Associative: $(f \circ g) \circ h = f \circ (g \circ h)$ .
Composites of one-one functions are one-one; composites of onto functions are onto.

ISC 2027 asks for composites of algebraic functions only.

The inner function acts first.

Different columns: $f \circ g \ne g \circ f$.
$x$	$g(x)$	$(f \circ g)(x)$	$f(x)$	$(g \circ f)(x)$
$0$	$2$	$3$	$-1$	$-1$
$1$	$5$	$24$	$0$	$2$
$2$	$8$	$63$	$3$	$11$

Formulas

Composite
$(f \circ g)(x) = f\big(g(x)\big)$

Key terms

Composite function: $f \circ g$ , defined by $x \mapsto f(g(x))$ .

6. Invertible functions and their graphs

$f : A \to B$ is invertible exactly when it is bijective; then $f^{-1}(y) = x \iff f(x) = y$ .

Method: check one-one and onto, write $y = f(x)$ , solve for $x$ , rename, and verify $f(f^{-1}(x)) = x$ . On a restricted domain (for example $x \ge 0$ ), the domain decides which root to keep.

The graph of $f^{-1}$ is the reflection of the graph of $f$ in $y = x$ ; a point where they meet lies on $y = x$ . Also $(f^{-1})^{-1} = f$ and $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ . Note that $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ .

$y = 3x - 2$
$y = \dfrac{x + 2}{3}$
$y = x$

A function and its inverse are mirror images in $y = x$.

Five steps, the last one a check.

$y = x^2 + 1$ , $x \ge 0$
$y = \sqrt{x - 1}$
$y = x$

Restricting the domain to $x \ge 0$ makes the square function invertible.

Formulas

Inverse of a composite
$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$

Key terms

Inverse function: $f^{-1}$ with $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$ .

Exam tips

How ISC marks are awarded

Equivalence proof (3 marks): one mark each for reflexive, symmetric and transitive, using general letters; classes earn further marks when asked.
Property fails (1–2 marks): one explicit counterexample.
Domain and range (2 marks): one each, with the reason.
Invertible and inverse (2–3 marks): one-one, onto, then the inverse; check by composing.
Section D (5 marks): questions come in parts such as 2 + 3; label each part.

Common mistakes

Reading $f \circ g$ as " $f$ first".
Writing $f^{-1}$ before showing $f$ is a bijection.
Keeping both roots on a restricted domain.
Forgetting the excluded value in the codomain of a rational function.

1. Relations on a set

Formulas

Key terms

2. Reflexive, symmetric, transitive and equivalence relations

Formulas

Key terms

3. Functions, domain and range

Key terms

4. One-one, many-one, onto and into functions

Formulas

Key terms

5. Composite functions

Formulas

Key terms

6. Invertible functions and their graphs

Formulas

Key terms

Exam tips

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