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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 $\{(a,b):a\in A,b\in B\}$.