Appendix A: Notation and Symbols

This is the canonical notation reference for the whole book. If a symbol ever looks unfamiliar, find it here. The conventions below are used consistently in every chapter.

A standing convention: in this book $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ — the natural numbers include 0. When a result needs the positive integers only, we write $\mathbb{Z}^{+}$ or say "$n \ge 1$."

Number systems

Symbol	Meaning
$\mathbb{N}$	Natural numbers $\{0, 1, 2, \dots\}$
$\mathbb{Z}$	Integers $\{\dots, -2, -1, 0, 1, 2, \dots\}$
$\mathbb{Z}^{+}$	Positive integers $\{1, 2, 3, \dots\}$
$\mathbb{Q}$	Rational numbers
$\mathbb{R}$	Real numbers
$\mathbb{Z}_n$	Integers modulo $n$, $\{0, 1, \dots, n-1\}$

Logic (Chapters 1–3)

Symbol	Meaning	Example
$\neg p$	not $p$ (negation)	$\neg \text{True} = \text{False}$
$p \land q$	$p$ and $q$ (conjunction)	true only if both hold
$p \lor q$	$p$ or $q$ (disjunction)	true if either holds
$p \oplus q$	$p$ xor $q$	true if exactly one holds
$p \rightarrow q$	$p$ implies $q$	false only when $p$ true, $q$ false
$p \leftrightarrow q$	$p$ if and only if $q$	true when $p,q$ have the same value
$p \equiv q$	$p$ is logically equivalent to $q$	same truth table
$\forall x\, P(x)$	for all $x$, $P(x)$	universal quantifier
$\exists x\, P(x)$	there exists $x$ with $P(x)$	existential quantifier
$\exists! x\, P(x)$	there exists a unique $x$
$\top,\ \bot$	true, false (tautology, contradiction)

Sets (Chapter 8, used throughout)

Symbol	Meaning
$x \in A$, $x \notin A$	$x$ is (is not) an element of $A$
$A \subseteq B$, $A \subset B$	$A$ is a subset, a proper subset, of $B$
$A \cup B$, $A \cap B$	union, intersection
$A \setminus B$, $\overline{A}$	set difference, complement
$\emptyset$	the empty set
$\mathcal{P}(S)$	power set (set of all subsets) of $S$
$\lvert S \rvert$	cardinality (size) of $S$
$A \times B$	Cartesian product
$\{x \mid P(x)\}$	set-builder: all $x$ such that $P(x)$

Functions and relations (Chapters 9, 12)

Symbol	Meaning
$f \colon A \to B$	$f$ is a function from $A$ to $B$
$g \circ f$	composition ($g$ after $f$)
$f^{-1}$	inverse function
$\lfloor x \rfloor$, $\lceil x \rceil$	floor, ceiling
$a \mathrel{R} b$	$a$ is related to $b$ under relation $R$
$[a]$	equivalence class of $a$
$\preceq$	a partial order

Number theory (Chapters 22–25)

Symbol	Meaning
$a \mid b$, $a \nmid b$	$a$ divides (does not divide) $b$
$\gcd(a,b)$, $\operatorname{lcm}(a,b)$	greatest common divisor, least common multiple
$a \equiv b \pmod{n}$	$a$ is congruent to $b$ modulo $n$
$a \bmod n$	the remainder of $a$ divided by $n$
$\phi(n)$	Euler's totient of $n$

Counting and probability (Chapters 15–21)

Symbol	Meaning
$n!$	$n$ factorial
$P(n,k)$	permutations of $n$ taken $k$ at a time
$\binom{n}{k}$	binomial coefficient ("$n$ choose $k$")
$\sum_{i=1}^{n} a_i$, $\prod_{i=1}^{n} a_i$	sum, product
$P(A)$, $P(A \mid B)$	probability of $A$; of $A$ given $B$
$E[X]$, $\operatorname{Var}(X)$	expected value, variance of $X$

Complexity (Chapter 14, used throughout)

Symbol	Meaning
$O(f)$	grows no faster than $f$ (upper bound)
$\Omega(f)$	grows no slower than $f$ (lower bound)
$\Theta(f)$	grows at the same rate as $f$ (tight bound)

Graphs (Chapters 27–34)

Symbol	Meaning
$G = (V, E)$	a graph with vertex set $V$ and edge set $E$
$\deg(v)$	degree of vertex $v$
$N(v)$	the neighborhood (adjacent vertices) of $v$
$K_n$	complete graph on $n$ vertices
$K_{m,n}$	complete bipartite graph
$\chi(G)$	chromatic number of $G$

Proof

Symbol	Meaning
$\blacksquare$	end of proof (QED)
$\therefore$	therefore
$\Rightarrow$, $\Leftrightarrow$	implies; if and only if (in reasoning)

A note for builders of this book: all mathematics here uses standard LaTeX commands (e.g., \mathbb{N}, \mathcal{P}, \gcd, \operatorname{lcm}, \pmod, \binom, \lceil), with $...$ for inline math and $$...$$ for displayed math. No custom macros are used, so the same source renders identically under Jupyter Book and mdBook.