Math Basics

Part 1 of 8 · Index · Next: Operators

How to read an unfamiliar formula

Read a formula in this order:

1. Find the main operator: $=$, $\le$, $\min$, $\sum$, $\Rightarrow$, etc.
2. Identify every variable and its domain.
3. Read subscripts and superscripts.
4. Read constraints written below operators.
5. Translate the formula into one plain-English sentence.

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1. Equality and comparison

Symbol	Name	Meaning
$a=b$	Equality	$a$ and $b$ have the same value
$a\ne b$	Not equal	$a$ and $b$ differ
$a:=b$	Definition	Define $a$ to be $b$
$a\equiv b$	Equivalence / identity	Equal by definition, always equal, or equivalent under a relation
$a\approx b$	Approximately equal	Values are close but not necessarily identical
$a\sim b$	Similar / asymptotically equivalent	Meaning depends on context
$a<b$	Strictly smaller	$a$ is smaller than $b$
$a\le b$	Smaller or equal	$a$ is at most $b$
$a>b$	Strictly greater	$a$ is greater than $b$
$a\ge b$	Greater or equal	$a$ is at least $b$
$a\propto b$	Proportional	$a=cb$ for some constant $c$

Common wording

$$x\le 10$$

Read:

$x$ is at most 10.

$$x\ge 10$$

Read:

$x$ is at least 10.

$$a:=b+1$$

Read:

Define $a$ as $b+1$.

The distinction between $=$ and $:=$ is useful:

$$x=5$$

asserts that $x$ has value 5.

$$x:=5$$

introduces or defines $x$ to have value 5.

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2. Number sets

Symbol	Name	Typical content
$\mathbb{N}$	Natural numbers	$0,1,2,\dots$, or sometimes $1,2,\dots$
${\mathbb{N}}_0$	Nonnegative integers	$0,1,2,\dots$
$\mathbb{Z}$	Integers	$\dots ,-2,-1,0,1,2,\dots$
$\mathbb{Q}$	Rational numbers	Fractions $a/b$
$\mathbb{R}$	Real numbers	All ordinary continuous numbers
${\mathbb{R}}_{\ge 0}$	Nonnegative reals	Real numbers $x\ge 0$
$\mathbb{C}$	Complex numbers	Numbers $a+bi$
$\{0,1\}$	Boolean/binary set	False/true or off/on

Warning

Some authors include zero in $\mathbb{N}$ and some do not. Check the author’s convention.

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3. Set notation

Symbol	Meaning
$x\in S$	$x$ belongs to set $S$
$x\notin S$	$x$ does not belong to $S$
$A\subseteq B$	Every element of $A$ belongs to $B$
$A⊊B$	$A$ is a strict subset of $B$
$\varnothing$	Empty set
$\{a,b,c\}$	Set containing $a,b,c$
$A\cup B$	Union: elements in $A$ or $B$
$A\cap B$	Intersection: elements in both
$A\setminus B$	Elements in $A$ but not in $B$
$A\times B$	Cartesian product
$|A|$	Number of elements in $A$
$\mathcal{P}(A)$ or $2^A$	Set of all subsets of $A$

Set-builder notation

$$\{x\in S:P(x)\}$$

Read:

The set of all $x$ belonging to $S$ such that $P(x)$ is true.

The colon means such that. A vertical bar is also common:

$$\{x\in S\mid P(x)\}$$

Example:

$$\{x\in \mathbb{N}:x<5\}=\{0,1,2,3,4\}$$

assuming zero belongs to $\mathbb{N}$.

Example from optimization:

$$T=\min \left\{t\in \mathbb{N}:C(t)\ge N\right\}$$

Read:

$T$ is the smallest natural number $t$ such that the capacity at time $t$ is at least $N$.

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4. Logical symbols

Symbol	Name	Meaning
$\neg P$	Negation	$P$ is false
$P\wedge Q$	Conjunction	$P$ and $Q$
$P\vee Q$	Disjunction	$P$ or $Q$, possibly both
$P\oplus Q$	Exclusive or	Exactly one of $P,Q$
$P\Rightarrow Q$	Implication	If $P$, then $Q$
$P\Leftarrow Q$	Reverse implication	$Q$ implies $P$
$P\iff Q$	Equivalence	$P$ if and only if $Q$
$\forall x$	Universal quantifier	For every $x$
$\exists x$	Existential quantifier	There exists an $x$
$\exists !x$	Unique existence	There exists exactly one $x$
$\therefore$	Therefore	The conclusion follows
$\because$	Because	Introduces a reason

Implication

$$P\Rightarrow Q$$

means:

Whenever $P$ is true, $Q$ must also be true.

It does not automatically mean:

$$Q\Rightarrow P$$

Example:

$$x>5\Rightarrow x>0$$

is true, but:

$$x>0\Rightarrow x>5$$

is false.

Equivalence

$$P\iff Q$$

means both:

$$P\Rightarrow Q$$

and:

$$Q\Rightarrow P$$

Read:

$P$ if and only if $Q$.

Often abbreviated iff.

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5. Quantifiers

Universal statement

$$\forall x\in S,P(x)$$

Read:

For every $x$ in $S$, property $P(x)$ holds.

Example:

$$\forall i\in \{1,\dots ,k\},d_i\ge 1$$

Read:

Every path length $d_i$ is at least 1.

Existential statement

$$\exists x\in S,P(x)$$

Read:

There is at least one $x$ in $S$ for which $P(x)$ holds.

Example:

$$\exists P\text{ from }s\text{ to }t$$

Read:

At least one path exists from $s$ to $t$.

Quantifier order matters

These statements are different:

$$\forall x,\exists y,P(x,y)$$

Every $x$ has some possibly different $y$.

$$\exists y,\forall x,P(x,y)$$

There is one single $y$ that works for every $x$.

Example:

$$\forall n\in \mathbb{N},\exists m\in \mathbb{N},m>n$$

is true.

But:

$$\exists m\in \mathbb{N},\forall n\in \mathbb{N},m>n$$

is false.

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6. Index notation

Subscript

$$x_i$$

usually means:

Element $i$ of a sequence, array, vector, or family.

Examples:

$$d_i=\text{length of path }i$$ $$x_i=\text{number of ants assigned to path }i$$

A double subscript:

$$a_{ij}$$

often means:

Entry at row $i$, column $j$ of a matrix.

Superscript

$$x^2$$

normally means exponentiation.

But:

$$x^{(k)}$$

often means the value of $x$ at iteration $k$, not $x$ raised to $k$.

Examples:

$$x^3=x\cdot x\cdot x$$

but:

$$x^{(3)}$$

may mean “the third version of $x$.”

Range notation

$$i=1,\dots ,k$$

means:

$$i\in \{1,2,\dots ,k\}$$

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