Sets, Relations and Functions Formula Sheet

Every must-know formula, identity, and counting result for Sets, Relations and Functions on a single scannable page. Use it for last-minute revision before JEE Main and Advanced.

Sets — Notation & Basics

Standard Number Sets

Symbol	Name	Elements
$\mathbb{N}$	Natural Numbers	$\{1, 2, 3, \dots\}$
$\mathbb{W}$	Whole Numbers	$\{0, 1, 2, 3, \dots\}$
$\mathbb{Z}$	Integers	$\{\dots, -2, -1, 0, 1, 2, \dots\}$
$\mathbb{Z}^+$	Positive Integers	Same as $\mathbb{N}$
$\mathbb{Q}$	Rationals	$\left\{\frac{p}{q} : p, q \in \mathbb{Z},\ q \neq 0\right\}$
$\mathbb{R}$	Reals	All rational and irrational numbers

Nesting of number sets:

$$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$

Set Membership & Equality

• $x \in A$ means $x$ belongs to $A$; $x \notin A$ means it does not.
• Set equality: $A = B \iff (x \in A \Leftrightarrow x \in B)$
• Subset: $A \subseteq B \iff (x \in A \Rightarrow x \in B)$
• Proper subset: $A \subset B \iff A \subseteq B$ and $A \neq B$
• Antisymmetry of $\subseteq$: if $A \subseteq B$ and $B \subseteq A$, then $A = B$

Subset Counting

For a set $A$ with $|A| = n$:

Quantity	Formula
Number of subsets	$2^n$
Number of proper subsets	$2^n - 1$
Number of non-empty subsets	$2^n - 1$
Number of non-empty proper subsets	$2^n - 2$

Interval Notation (Real Numbers)

Interval	Set-Builder
$[a, b]$	$\{x : a \leq x \leq b\}$
$(a, b)$	$\{x : a < x < b\}$
$[a, b)$	$\{x : a \leq x < b\}$
$(a, b]$	$\{x : a < x \leq b\}$
$[a, \infty)$	$\{x : x \geq a\}$
$(-\infty, b)$	$\{x : x < b\}$

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Set Operations

Core Operations

$$\boxed{A \cup B = \{x : x \in A \text{ or } x \in B\}}$$$$\boxed{A \cap B = \{x : x \in A \text{ and } x \in B\}}$$$$\boxed{A - B = \{x : x \in A \text{ and } x \notin B\}}$$$$\boxed{A' = \{x : x \in U \text{ and } x \notin A\} = U - A}$$

Disjoint sets: $A \cap B = \emptyset$.

Symmetric difference:

$$A \triangle B = (A - B) \cup (B - A) = (A \cup B) - (A \cap B)$$

Complement Properties

Property	Result
Double complement	$(A')' = A$
Exhaustive	$A \cup A' = U$
Mutually exclusive	$A \cap A' = \emptyset$
Universal complement	$U' = \emptyset$
Empty complement	$\emptyset' = U$

Algebraic Laws

Law	Statement
Commutative	$A \cup B = B \cup A$,  $A \cap B = B \cap A$
Associative	$(A \cup B) \cup C = A \cup (B \cup C)$,  $(A \cap B) \cap C = A \cap (B \cap C)$
Distributive	$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
Distributive	$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
Identity	$A \cup \emptyset = A$,  $A \cap U = A$
Domination	$A \cup U = U$,  $A \cap \emptyset = \emptyset$
Idempotent	$A \cup A = A$,  $A \cap A = A$
Absorption	$A \cup (A \cap B) = A$,  $A \cap (A \cup B) = A$

De Morgan’s Laws

$$\boxed{(A \cup B)' = A' \cap B'} \qquad \boxed{(A \cap B)' = A' \cup B'}$$

Generalized form:

$$(A_1 \cup A_2 \cup \dots \cup A_n)' = A_1' \cap A_2' \cap \dots \cap A_n'$$$$(A_1 \cap A_2 \cap \dots \cap A_n)' = A_1' \cup A_2' \cup \dots \cup A_n'$$

Power Set

The power set $P(A)$ is the set of all subsets of $A$: $P(A) = \{S : S \subseteq A\}$.

$$\boxed{|P(A)| = 2^{|A|} = 2^n}$$

Properties: $\emptyset \in P(A)$ and $A \in P(A)$ for any set $A$; if $A \subseteq B$ then $P(A) \subseteq P(B)$.

Cardinality (Inclusion–Exclusion)

Two sets:

$$\boxed{|A \cup B| = |A| + |B| - |A \cap B|}$$

Three sets:

$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |A \cap C| + |A \cap B \cap C|$$

General principle for $n$ sets:

$$\left|\bigcup_{i=1}^{n} A_i\right| = \sum|A_i| - \sum|A_i \cap A_j| + \sum|A_i \cap A_j \cap A_k| - \dots + (-1)^{n+1}|A_1 \cap \dots \cap A_n|$$

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Relations

Cartesian Product & Relations

$$\boxed{A \times B = \{(a, b) : a \in A \text{ and } b \in B\}}$$

• Ordered-pair equality: $(a, b) = (c, d) \iff a = c$ and $b = d$
• Cardinality: $|A \times B| = |A| \times |B|$
• Empty product: $A \times \emptyset = \emptyset = \emptyset \times A$

A relation $R$ from $A$ to $B$ is any subset $R \subseteq A \times B$. If $(a,b) \in R$ we write $aRb$.

Domain, Co-domain, Range

Term	Meaning
Domain $\text{Dom}(R)$	Set of first elements of the ordered pairs in $R$
Co-domain	The whole target set $B$
Range $\text{Range}(R)$	Set of second elements of the ordered pairs in $R$

$$\boxed{\text{Range}(R) \subseteq \text{Co-domain}}$$

Inverse of a Relation

$$R^{-1} = \{(b, a) : (a, b) \in R\}$$

• $(R^{-1})^{-1} = R$
• $\text{Dom}(R^{-1}) = \text{Range}(R)$ and $\text{Range}(R^{-1}) = \text{Dom}(R)$

Counting Relations

For $|A| = m$ and $|B| = n$ (so $|A \times B| = mn$):

Quantity	Formula
Number of relations from $A$ to $B$	$2^{mn}$
Empty relation	$1$
Universal relation	$1$
Non-empty relations	$2^{mn} - 1$

For a relation on a set $A$ with $|A| = n$:

Type	Count
Reflexive relations	$2^{n(n-1)}$
Symmetric relations	$2^{\frac{n(n+1)}{2}}$
Reflexive and symmetric relations	$2^{\frac{n(n-1)}{2}}$

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Types of Relations

Defining Properties (relation $R$ on set $A$)

$$\boxed{\text{Reflexive: } (a, a) \in R \ \text{ for all } a \in A}$$$$\boxed{\text{Symmetric: } (a, b) \in R \Rightarrow (b, a) \in R}$$$$\boxed{\text{Anti-symmetric: } (a, b) \in R \text{ and } (b, a) \in R \Rightarrow a = b}$$$$\boxed{\text{Transitive: } (a, b) \in R \text{ and } (b, c) \in R \Rightarrow (a, c) \in R}$$

Property	Matrix check
Reflexive	All diagonal entries are $1$
Symmetric	$M = M^{T}$ (matrix equals its transpose)
Transitive	Every chain $a \to b \to c$ closes with $a \to c$

Equivalence Relation

$$\boxed{\text{Equivalence} = \text{Reflexive} + \text{Symmetric} + \text{Transitive}}$$

Equivalence class of $a$:

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

Properties of equivalence classes:

• Every element lies in exactly one class.
• Any two classes are either identical or disjoint.
• The classes partition the set $A$ (non-overlapping, union is all of $A$).

Standard examples: equality $=$, congruence $a \equiv b \pmod{n}$, parallel lines $\parallel$.

Quick Comparison

Relation	Reflexive	Symmetric	Transitive	Equivalence
$=$	Yes	Yes	Yes	Yes
$\leq$	Yes	No	Yes	No
$<$	No	No	Yes	No
$\perp$ (perpendicular)	No	Yes	No	No
$\parallel$ (parallel)	Yes	Yes	Yes	Yes

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Functions — Basics

A function $f: A \to B$ is a relation in which every element of $A$ is related to exactly one element of $B$.

$$\boxed{\text{Range}(f) \subseteq \text{Co-domain}}$$

Terminology

Term	Meaning
Domain $\text{Dom}(f)$	Set of all valid inputs (set $A$)
Co-domain	Set of possible outputs (set $B$)
Range	Set of actual outputs
Image	$f(x)$, the output of a specific input
Pre-image	Input(s) giving a specific output, $f^{-1}(\{y\})$

Domain Restrictions (most common)

Expression	Restriction
$\dfrac{1}{g(x)}$	$g(x) \neq 0$
$\sqrt{g(x)}$	$g(x) \geq 0$
$\log\,(g(x))$	$g(x) > 0$
$\tan x$	$x \neq \dfrac{\pi}{2} + n\pi$

Standard Functions

Function	Definition	Range
Identity	$I(x) = x$	$A$
Constant	$f(x) = c$	$\{c\}$
Modulus	$f(x) = \lvert x \rvert$	$[0, \infty)$
Greatest integer (floor)	$f(x) = \lfloor x \rfloor$ = greatest integer $\leq x$	integers
Fractional part	$\{x\} = x - \lfloor x \rfloor$	$[0, 1)$
Signum	$\text{sgn}(x) = 1,\,0,\,-1$ for $x>0,\,x=0,\,x<0$	$\{-1, 0, 1\}$

Algebra of Functions

For $f, g: A \to \mathbb{R}$:

Operation	Definition	Domain
$(f + g)(x)$	$f(x) + g(x)$	$\text{Dom}(f) \cap \text{Dom}(g)$
$(f - g)(x)$	$f(x) - g(x)$	$\text{Dom}(f) \cap \text{Dom}(g)$
$(fg)(x)$	$f(x) \cdot g(x)$	$\text{Dom}(f) \cap \text{Dom}(g)$
$(f/g)(x)$	$\dfrac{f(x)}{g(x)}$	$\text{Dom}(f) \cap \text{Dom}(g) \cap \{x : g(x) \neq 0\}$

Even & Odd Functions

• Even: $f(-x) = f(x)$ — graph symmetric about the $y$-axis (e.g. $x^2,\ \cos x,\ |x|$).
• Odd: $f(-x) = -f(x)$ — graph symmetric about the origin (e.g. $x^3,\ \sin x,\ x$).

Any function splits into even + odd parts:

$$f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even}} + \underbrace{\frac{f(x) - f(-x)}{2}}_{\text{odd}}$$

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Types of Functions

Definitions

$$\boxed{\text{One-one (injective): } f(x_1) = f(x_2) \Rightarrow x_1 = x_2}$$$$\boxed{\text{Onto (surjective): } \text{Range}(f) = \text{Co-domain } B}$$$$\boxed{\text{Bijective} = \text{Injective} + \text{Surjective}}$$

Checking one-one for differentiable $f$: strictly monotonic works — $f'(x) > 0$ everywhere (increasing) or $f'(x) < 0$ everywhere (decreasing) implies one-one.

Counting Functions

For $|A| = m$ and $|B| = n$:

Type	Formula	Condition
All functions	$n^m$	Always
One-one (injective)	$\dfrac{n!}{(n-m)!} = P(n, m)$	$m \leq n$
Onto (surjective)	$\displaystyle\sum_{r=0}^{n-1} (-1)^r \binom{n}{r}(n-r)^m$	$m \geq n$
Bijective	$n!$	$m = n$

Small onto cases worth memorizing:

$$\text{Onto } m \to 2:\ \ 2^m - 2 \qquad\qquad \text{Onto } m \to 3:\ \ 3^m - 3\cdot 2^m + 3$$

graph TD
    F[Functions] --> O["One-One (Injective)"]
    F --> ON["Onto (Surjective)"]
    F --> B[Bijective]
    O -->|"Different inputs → different outputs"| O1["|A| ≤ |B|"]
    ON -->|"Range = Co-domain"| ON1["|A| ≥ |B|"]
    B -->|"One-One + Onto"| B1["|A| = |B|, has inverse"]

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Composition & Inverse Functions

Composition

For $f: A \to B$ and $g: B \to C$:

$$\boxed{(g \circ f)(x) = g(f(x))}$$

Domain: $\text{Dom}(g \circ f) = \{x \in A : f(x) \in \text{Dom}(g)\}$.

Property	Statement
Not commutative	$g \circ f \neq f \circ g$ in general
Associative	$(h \circ g) \circ f = h \circ (g \circ f)$
Identity	$f \circ I = I \circ f = f$

Preservation under composition:

If	Then
$f, g$ one-one	$g \circ f$ one-one
$f, g$ onto	$g \circ f$ onto
$f, g$ bijective	$g \circ f$ bijective
$g \circ f$ one-one	$f$ is one-one
$g \circ f$ onto	$g$ is onto

Inverse Functions

For a bijective $f: A \to B$, the inverse $f^{-1}: B \to A$ satisfies:

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

Property	Statement
Left inverse	$f^{-1} \circ f = I_A$
Right inverse	$f \circ f^{-1} = I_B$
Involutive	$(f^{-1})^{-1} = f$
Reverses order	$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$
Derivative of inverse	$(f^{-1})'(y) = \dfrac{1}{f'(x)}$ where $y = f(x)$

• Graph of $f^{-1}$ is the reflection of $f$ about the line $y = x$: if $(a,b)$ lies on $f$, then $(b,a)$ lies on $f^{-1}$.
• Self-inverse (involution): $f(f(x)) = x$, i.e. $f^{-1} = f$. Examples: $f(x) = \frac{1}{x}$, $f(x) = -x$, $f(x) = \frac{a - x}{1 + ax}$.
• If $f \circ g = I$ and $g \circ f = I$, then $f$ and $g$ are inverses of each other.

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One-Glance Recap

Concept	Headline formula
Subsets of $A$	$2^n$
Power set size	$\lvert P(A)\rvert = 2^n$
Two-set count	$\lvert A \cup B\rvert = \lvert A\rvert + \lvert B\rvert - \lvert A \cap B\rvert$
De Morgan	$(A \cup B)' = A' \cap B'$,  $(A \cap B)' = A' \cup B'$
Relations $A \to B$	$2^{mn}$
Reflexive relations	$2^{n(n-1)}$
Symmetric relations	$2^{n(n+1)/2}$
Total functions	$n^m$
One-one functions	$\dfrac{n!}{(n-m)!}$
Bijections ($m=n$)	$n!$
Composition	$(g \circ f)(x) = g(f(x))$
Inverse of composition	$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$

Related: Sets and Their Representation · Set Operations · Introduction to Relations · Types of Relations · Introduction to Functions · Types of Functions · Composition of Functions