Notation

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# Notation and Abbreviations

## Proof notation

The following notation is commonly used in proofs:

• $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$ for the sets of natural numbers, integers, rationals, and reals, respectively. $\mathbb{C}$ is used for complex numbers;
• $\in$, as in $x\in A$ to indicate that an element is contained in a set [you should not use a curly epsilon $\varepsilon$];
• $\Rightarrow$ meaning implies, and $\Leftarrow$ meaning only if;
• $\Leftrightarrow$ meaning if and only if or is equivalent to;
• $\{n\in\mathbb{Z}: n \text{ is even}\}$ or $\{n\in\mathbb{Z}| n \text{ is even}\}$to indicate a set constructed of all elements with a given property. Here the colon $:$ is read as "such that";
• $f:A\to B$ to indicate a particular function's domain and range, read as "$f$ takes $A$ to $B$";

The following abbreviations are also commonly used:

• BWOC: by way of contradiction;
• WTS: want to show;
• QED: quad erat demonstratum or that which was to be shown;
• IFF: if and only if;
• WLOG: without loss of generality.

The following notations are less commonly used; I do not advise you to use them very often in proofs:

• $\forall$ meaning "for each" or "for all";
• $\exists$ meaning "there exists", and $\nexists$ meaning "there does not exist";
• $\therefore$ meaning "therefore";
• $\ni$ or $|$ or $:$ meaning "such that" ($|$ and $:$ are usually used only in set notation);

## More Notation

These notations are specific to particular mathematical structures (e.g. logic, equivalence relations)

Equivalence relations
$\sim$ means equivalent;
Logic
$\wedge$ means "and", $\vee$ means "or", and $\lnot$ means "not";
Set Theory
$\emptyset$ represents the empty set, $A\cap B$ represents the intersection of sets, $A\cup B$ represents the union of sets; $A^c$ represents the complement of a set; $A+A'$ represents the set of all elements which can be written as the sum of an element in $A$ and an element in $A'$; $A*A'$, $\frac{1}{A}$, etc. are defined similarly.
Cuts
$A|B$ indicates a cut.
Inner products
$\langle \vec x,\vec y\rangle$ and $\vec x \cdot \vec y$ both indicate the dot product or inner product of two vectors
Euclidean space
$(a,b)$ indicates the open interval $\{x:a<x<b\}$ of the real numbers and $[a,b]$ represents the closed interval $\{x:a\leq x\leq b\}$; $\mathbb{R}^n$ represents the set of n-vectors of the form $(x_1,\ldots,x_n)$; the Cartesian product $A\times B$ of sets $A\subset\mathbb{R}$ and $B\subset\mathbb{R}$ is the set of vectors $\{(a,b):a\in A,b\in B\}$.