See also: [[LaTeX]]
> The symbol denotes whatever the author tells you it will denote in his comments about notation, and there is a special place in hell for users of unexplained notation.
\- StackOverflow [comment](https://mathoverflow.net/a/11496) by Mariano Suárez-Álvarez
# Structures
Algebraic convention uses letters near the beginning of the alphabet for given values and letters near the end for unknown ones.
Coordinates of points are considered unknown by default, perhaps because they are often the values that geometric problems are attempting to solve for.
Thus, anything coordinate-like tends to use $x, y, z$. Sets and line segments are typically dealing with concrete values, so they use $A, B, C$.
| Notation | Reading | Info |
| ------------------------------------------------ | ---------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| $A$ | *set* reference | italic capital letters are used to indicate sets |
| $\set{x,y,z}$ | *set contents* | curly brackets around comma separated values |
| $\vec v$ | *vector* reference | lowercase italic letters with an arrow pointing right indicate vectors, often using $v$ to start |
| $\begin{bmatrix}x\; y\; z\end{bmatrix}$ | *vector contents* (row) | - square brackets around space separated values<br> - I don't use this comma-less syntax, so that matrices and vectors are distinct, but still share common brackets to show how they are related |
| $\begin{bmatrix}x, y, z\end{bmatrix}$ | *vector contents* (row - alternate) | square brackets around comma separated values |
| ${\begin{bmatrix}x\\y\\\vdots \\z\end{bmatrix}}$ | *vector contents* (column - standard notation) | - tall square brackets around space separated vertical values<br> - mostly used when treating a vector as a single column matrix<br> - in fact the way vector contents are specified in TeX is using matrix syntax |
| $\begin{bmatrix}x; y; z\end{bmatrix}$ | *vector contents* (column - alternate) | square brackets around semicolon separated values |
| $\begin{bmatrix}x\; y\; z\end{bmatrix}^T$ | *vector contents* (column - transform) | column vectors may also be written identically to row vectors but with a "transform" superscript capital T |
| $\vec v_x$ | *vector components* (basis) | - subscript indicates the basis component, unit, or field of the vector by name<br> - used most often with finite fixed length vectors<br> - there are some papers which use superscript components, but I don't like the ambiguity with power notation |
| $v_1$ | *vector components* (index) | - the individual components of a vector may be indicated by using the name of the vector with a subscript number to indicate their position<br> - used most often with variable, ambivalent, or unknown length vectors<br> - this form is often used within the vector contents notation when describing the structure of the vector and in relation to matrices |
| $(x, y, z)$ | *coordinates* | parentheses around comma separated values |
| $\mathbf A$ | *matrix* reference (bold) | bold capital letters are used to indicate matricies |
| $\underline{\underline{\mathbf A}}$ | *matrix* reference (double underline) | - double underline capital (bold or not)<br> - useful when writing by hand to better distinguish a matrix from a set |
| $\begin{bmatrix}a_1&a_2\\ a_3&a_4\end{bmatrix}$ | *matrix contents* (square) | - square brackets around space separated rows and columns of values<br> - note that `\,`, `\:`, and `\;` insert spaces of a fixed width *within* an element, while `&` *separates* elements while keeping them perfectly aligned in columns |
| $\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}$ | *matrix contents* (round) | - parentheses around space separated rows and columns of values<br> - same as the square variant<br> - I don't like this because parens are used for grouping and coordinates which operate very differently, while the square brackets shared with vectors are |
| $a_{1,2}$ | *matrix entry* (miniscule) | - variable entries in a matrix are often written as a lowercase version of the matrix name with a subscript numeric position |
| $\mathbf A_{1,2}$ | *matrix entry* (majuscule) | - however, subscripting the capital is fine too, particularly outside of the contents definition<br> - personally I would prefer to continue using the same symbol for consistency rather than a lowercase version of it, but I definitely understand the appeal of being able to tell at a glance if we're talking about the whole thing or just a single entry |
# Constants
## Zero Elements
| Notation | Reading | Info |
| ------------- | ------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| $O$ | *origin* | graph *origin* for a coordinate system of any size<br> - $(0,0)$ in a standard 2-dimensional coordinate system<br> - italic capital "O" |
| $O_{m,n}$ | *zero matrix* | null matrix for a matrix of any size<br> - $\begin{bmatrix}0\; 0 \\ 0\; 0\end{bmatrix}$ in a simple 2x2 matrix<br> - identical to the graph origin character except that it may show the dimensions in a subscript<br> - it seems to me that a bold capital $\mathbf O_{m,n}$ or $\mathbf 0_{m,n}$ |
| ${\vec {0}}$ | *zero vector* | $\begin{bmatrix}0, 0, 0\end{bmatrix}$ in a standard 3-dimensional basis |
| $\varnothing$ | *empty set* | also:<br> - $\set{}
lt;br> - $\emptyset$ |
# Logic
## Equivalence
The below operations are often used interchangeably for a variety of reasons and in different contexts. The guidelines below govern how I would default to reading and using them.
These all indicate the *if-and-only-if* operation which is the statement that "either both A and B are true or both are false".
| Notation | Reading | Info |
| --------------------- | ---------------------- | --------------------- |
| $A \leftrightarrow B$ | *material equivalence* | for the current model |
| $A \iff B$ | *logical equivalence* | in every model |
| $A \equiv B$ | *equivalent* | |
## Partial Equivalence
| Notation | Reading | Info |
| ----------- | -------------------------- | --------------------------- |
| $\leq$ | *less than or equal to* | |
| $\leqslant$ | *less than or equal to* | |
| $\geq$ | *greater than or equal to* | |
| $\geqslant$ | *greater than or equal to* | |
| $\lt$ | *less than* | |
| $\gt$ | *greater than* | |
| $\ll$ | *much less than* | orders of magnitude less |
| $\gg$ | *much greater than* | orders of magnitude greater |
| $\preceq$ | *at most* | |
| $\succeq$ | *at least* | |
## Similarity and Inequality
| Notation | Reading | Info |
| -------------- | --------------------- | ---------------------------------- |
| $\sim$ | *similar* | within the same order of magnitude |
| $\thicksim$ | *similar* | "thick" |
| $\nsim$ | *not similar* | |
| $\approx$ | *approximately* | almost equal, a simplified form of |
| $\thickapprox$ | *approximately* | "thick" |
| $\approxeq$ | *approximately equal* | |
| $\cong$ | *congruent* | |
| $\ncong$ | *not congruent* | |
| $\simeq$ | *similar equal* | |
| $\neq$ | *not equal* | |
## And
These all indicate the *logical conjunction* operation which is the statement that "both A and B must be true, otherwise false".
| Notation | Reading | Info |
| ----------- | ------------------------ | -------------------------------------------------------------------------------------------------------------- |
| $A \land B$ | *logical and* | To avoid confusion with wedge products (eg of vectors), I **avoid** this notation where possible |
| $A \cdot B$ | *logical multiplication* | Makes sense for **numerical** logic where multiplication gives zero (false) or non-zero (true) results |
| $A\;\&\;B$ | *and* | This is most similar to the way it is written in English and in most programming languages, so I **prefer** it |
## Or
These all indicate the *logical inclusive disjunction* operation which is the statement that "at least one of A or B must be true, otherwise false".
| Notation | Reading | Info |
| --------------- | ------------------------ | -------------------------------------------------------------------------------------------------------------- |
| $A \lor B$ | *logical or* | This pairs with the "$\wedgequot; but is not its opposite, I **avoid** this notation where possible |
| $A + B$ | *logical multiplication* | Makes sense for **numerical** logic where addition gives zero (false) or non-zero (true) results |
| $A \parallel B$ | *or* | This is most similar to the way it is written in English and in most programming languages, so I **prefer** it |
## Not
These all indicate the *negation* operation which is the statement that "if A is true then false, otherwise true".
| Notation | Reading | Info |
| -------- | --------- | --------------------------------------------------------------------------------------------------- |
| $\neg A$ | *negate* | |
| $\not A$ | *slashed* | |
| $\sim A$ | | This is the same symbol as the one used for similarity, definitely **avoid** this wherever possible |
## Proofs
| Notation | Reading | Info |
| ---------------- | ------------ | -------------------------------------------------- |
| $A \therefore B$ | *therefore* | the state of $B$ is the logical consequence of $A$ |
| $A \because B$ | *because* | the state of $A$ is the logical consequence of $B$ |
| $A \implies B$ | *implies* | the state of $B$ is implied by $A$ |
| $A \impliedby B$ | *implied by* | the state of $A$ is implied by $B$ |
## Modal Logic
### Essential Operators
The common $\Box$ and $\lozenge$ operators are used in modal logic to indicate the more essential and the less essential, respectively. These operators are used and reused in nearly every form of modal logic to represent their two basic operations.
| Notation | Reading | Info |
| ---------- | ------------- | ----------------- |
| $\Box$ | *necessarily* | alethic logic |
| $\Diamond$ | *possibly* | alethic logic |
| $\Box$ | *provable* | provability logic |
| $\Diamond$ | *consistent* | provability logic |
| $\Box$ | *obligatory* | deontic logic |
| $\Diamond$ | *permissible* | deontic logic |
| $\Box$ | *always* | temporal logic |
| $\Diamond$ | *eventually* | temporal logic |
### Letter Symbols and Additional Operators
| Notation | Reading | Info |
| ------------------ | --------------- | ----------------------- |
| $\boldsymbol O B$ | *obligatory* | deontic logic (abbr) |
| $\boldsymbol P E$ | *permissible* | deontic logic (abbr) |
| $\boldsymbol I M$ | *impermissible* | deontic logic (abbr) |
| $\boldsymbol O M$ | *omissible* | deontic logic (abbr) |
| $\boldsymbol O P$ | *optional* | deontic logic (abbr) |
| $\boldsymbol N O$ | *non-optional* | deontic logic (abbr) |
| $O$ | *obligatory* | deontic logic (letter) |
| $P$ | *permissible* | deontic logic (letter) |
| $F$ | *forbidden* | deontic logic (letter) |
| $G$ | *will always* | temporal logic (letter) |
| $F$ | *future* | temporal logic (letter) |
| $H$ | *has always* | temporal logic (letter) |
| $P$ | *past* | temporal logic (letter) |
### Obscure Operators
I haven't found a good reference to these, but there are a handful of examples out there. These are primarily guesswork.
| Notation | Reading | Info |
| ------------------ | ------------ | ---- |
| $\bigcirc$ | *next* | |
| $\bigtriangleup$ | *non-con..?* | |
| $\bigtriangledown$ | *con..?* | |
| $\rhd$ | *future* | |
| $\lhd$ | *past* | |
## Others
These are the readings used in"boolean" logic.
| Notation | Reading | Info |
| --------------------- | ------- | ------------------------------------------------------------------------------------------------------------------ |
| $A \oplus B$ | *xor* | if either $A$ or $B$ are true but not both |
| $A \overline \lor B$ | *nor* | neither $A$ nor $B$ are true |
| $A \overline \land B$ | *nand* | both $A$ and $B$ are false |
| $A \odot B$ | *xnor* | either both are true or both or false<br>similar to if-and-only-if, but specifically represents the xnor operation |
# Numeric Operations
| Notation | Reading | Info |
| ------------ | ------------------------------------------- | ---------------------------- |
| $AB$ | numeric multiplication | Default operation for domain |
| $A \cdot B$ | numeric multiplication, vector dot product, | |
| $A \wedge B$ | vector wedge product | |
| | | |
# Attributes
| Notation | Reading | Info |
| --------------------------------- | ------------------- | ----------------------------------------- |
| $\lvert\lvert \vec v\rvert\rvert$ | *magnitude* | distance (length) of a vector from origin |
| $\lvert x\rvert$ | *absolute value* | distance of of a scalar from zero |
| $f'$ | *derivative* | derivative of a function |
| $f''$ | *second derivative* | second derivative of a function |
# References
## Formatting
### Helpers
- https://wumbo.net
- http://detexify.kirelabs.org/classify.html
- https://www.atomurl.net/math/
- http://asciimath.org/
### Documentation
- https://help.obsidian.md/Editing+and+formatting/Advanced+formatting+syntax
- https://en.wikibooks.org/wiki/LaTeX/Mathematics
- https://mirrors.rit.edu/CTAN/info/symbols/comprehensive/symbols-a4.pdf
- https://docs.mathjax.org/en/latest/basic/mathjax.html
- https://docs.mathjax.org/en/latest/input/tex/extensions/index.html
- https://www.tug.org/teTeX/tetex-texmfdist/doc/latex/amsmath/amsldoc.pdf
- https://www.ctan.org/tex-archive/info/symbols/comprehensive/?lang=en
### Sets
- https://tex.stackexchange.com/questions/253077/how-do-you-create-a-set-in-latex
### Vectors
- https://tex.stackexchange.com/questions/396147/row-vector-problem
- https://www.physicsread.com/latex-vector-arrow/
## Notation
Using browser inspection tools, Wikipedia can be made to expose the TeX syntax for any of its displayed equations or mathematical characters.
- https://en.wikipedia.org/wiki/List_of_mathematical_symbols_by_subject
- https://web.archive.org/web/20180619212427/ftp://ftp.ams.org/pub/tex/doc/amsmath/amsldoc.pdf
- https://github.com/igorsvara/Latex-Symbols
### Conventions
- https://cims.nyu.edu/~tjl8195/survey/results.html
### Coordinates
- https://en.wikipedia.org/wiki/Cartesian_coordinate_system?useskin=vector#Notations_and_conventions
### Vectors & Matricies
- https://en.wikipedia.org/wiki/Vector_notation
- https://en.wikipedia.org/wiki/Row_and_column_vectors
- https://math.stackexchange.com/questions/552347/notation-subscript-vs-superscript-for-coordinate-vector-fields
- https://en.wikipedia.org/wiki/Matrix_(mathematics)
- Labeling rows & columns of matrices and array tables
- https://tex.stackexchange.com/questions/30791/array-with-labeling-columns
- https://tex.stackexchange.com/questions/223501/matrix-with-rows-and-columns-labeled
- https://tex.stackexchange.com/questions/59517/label-rows-of-a-matrix-by-characters
- https://math.stackexchange.com/questions/307353/how-do-you-write-represent-the-all-ones-matrix
### Zero Elements
- https://en.wikipedia.org/w/index.php?title=Zero_element
- https://en.wikipedia.org/wiki/Zero_matrix
### Logic
- https://en.wikipedia.org/wiki/List_of_logic_symbols
- https://www.actual.world/resources/tex/doc/Modals.pdf
- https://bd.openlogicproject.org/
- https://latexref.xyz/Math-symbols.html
- https://tex.stackexchange.com/questions/695242/consistent-way-to-typeset-modality-symbols
### Equality and Inference
- https://www.geeksforgeeks.org/equality-and-inference-symbols-in-latex/