# Help:Editing/House Style

This page lists the various house style conventions that have been adopted on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Active contributors are expected to gradually master these, but when you first start to contribute, seasoned editors will come in and tidy the pages you write or create.

You are encouraged to study the changes made, and hence to pick up the style as you progress.

## Linking

Due to the desired standard of rigor on $\mathsf{Pr} \infty \mathsf{fWiki}$, there are a lot of concepts on any given (proof) page that have their own, dedicated Proof or Definition page on $\mathsf{Pr} \infty \mathsf{fWiki}$.

To ensure ease of reference and maximal clarity and consistency, the following rules for internal reference are to be adhered to.

For information on creating links, see this section.

### References to Theorems and Axioms

*Whenever* a theorem is invoked or referred to, be it in a proof or, for example, a clarifying comment, it should be referenced by its full title.

Also, for ease of editing, there is no need to change the case of theorem names; the $\mathsf{Pr} \infty \mathsf{fWiki}$ page title will suffice.

Thus, for example, a valid reference to the result Union Distributes over Intersection is simply:

- "By Union Distributes over Intersection, $A \cup \left({B \cap C}\right) = \left({A \cup B}\right) \cap \left({A \cup C}\right)$."

This is achieved by simply putting the title of the page you want to reference between double square brackets, `[[`

and `]]`

.

The same convention applies to axioms, except that the namespace identifier **Axiom:** should be removed.

The correct way to reference the page Axiom:Axiom of Choice thus is:

which is produced by:

`[[Axiom:Axiom of Choice|Axiom of Choice]]`

### References to Definitions

Whenever a concept (or part of it) is invoked on a page, it should be presented as a link to the definition of that concept.

It is preferred that **every** use of the word for that concept is presented as such a link.

These references are made in a non-intrusive way. Thus, we write:

- Let $R$ be a ring.

and not:

- Let $R$ be a Ring (Abstract Algebra).

### Permanent Redirects to Definitions on Subpages

Many definitions have subpages, for example:

In such cases there is (or **ought** to be) a **permanent redirect** to such a page, which is to be used instead in all cases.

In this case we have:

If you find you need to link to such a subpage, use the "What links here" (under "Tools" in the menu on the left hand side) to find out what permanent redirect may be available.

If there is none, feel free to implement one.

- Note

**This of course does not apply when the subpage is to handle multiple definitions**, for example:

- Definition:Prime Number/Definition 1
- Definition:Prime Number/Definition 2
- Definition:Prime Number/Definition 3

... and so on.

The rule here is (unless there are specific reasons to invoke exactly that instance of the definition, either to avoid circular arguments or in proofs of the equivalence of such definitions) **not** to invoke the individual subpage, but merely the top level page (in this case Definition:Prime Number). Otherwise maintenance becomes significantly more difficult.

## Mathematical Symbols

### Symbol Set

The only symbols that are accepted in $\mathsf{Pr} \infty \mathsf{fWiki}$ $\LaTeX$ source code are the standard alphanumeric and punctuation characters that can be found on a standard English-language keyboard. Letters with diacritical marks such as "á" should not be used.

If non-English characters are needed within $\LaTeX$ source code (the obvious instance being Greek), then the full $\LaTeX$ definition is to be used.

For example: $\alpha$ is to be rendered by the code `$\alpha$`

.

The same applies to specialised mathematical symbols. While it is appreciated that some contributors may have favourite techniques to allow them to place various mathematical symbols directly into the wiki text, such techniques are not portable and cause rendering issues in some browsers.

The only exceptions to this rule are:

- when reporting the name of a mathematician, for example: Wacław Franciszek Sierpiński
- when explaining the linguistic derivation of a term in, for example, a Language Note section. It is standard, for example, to use Greek characters directly here, rather than their $\LaTeX$ codes.

### Inline Equations

Inline equations (that is, those that appear as part of a text sentence) merely need the dollar delimiters. For example:

- The semilinear wave equation $\partial_t U = A U + \map B U$ is Hamiltonian.

is produced by the input:

- The semilinear wave equation
`$\partial_t U = A U + \map B U$`

is Hamiltonian.

### Displayed Equations

Displayed equations should be indented using a single colon, for example, a displayed equation should look like:

- $\ds \map H U = \int_0^{2 \pi} \frac {\paren {\partial_x u}^2} 2 + \frac {v^2} 2 - \map F u \rd x$

which you can enter as:

`:$\ds \map H U = \int_0^{2 \pi} \frac {\paren {\partial_x u}^2} 2 + \frac {v^2} 2 - \map F u \rd x$`

Note the $\mathsf{Pr} \infty \mathsf{fWiki}$ custom command `\ds`

which is a convenience abbreviation for the full command `\displaystyle`

.

Using a format that places the equation on the center of the page:

- $$E = m c^2$$

is discouraged, because with our "short sentence" house style, this breaks up the reading flow.

### Big Operators

The `\ds`

command should be used at the front of expressions using the 'big operators' such as `\sum`

and `\prod`

, whether the equation is **displayed** or **inline**.

This includes (but may not be exclusive to) the commands `\frac`

, `\binom`

, `\lim`

, `\bigcup`

, `\bigcap`

, `\int`

, `\sum`

and `\prod`

.

For example:

- $\sum_{i \mathop = 1}^n$
- $\prod_{i \mathop = 1}^n$
- $\frac {-b \pm \sqrt {b^2 - 4ac} } {2 a}$
- $\lim_{n \to \infty} \frac 1 n$

all look better as:

- $\ds \sum_{i \mathop = 1}^n$
- $\ds \prod_{i \mathop = 1}^n$
- $\ds \frac {-b \pm \sqrt {b^2 - 4ac} } {2 a}$
- $\ds \lim_{n \mathop \to \infty} \frac 1 n$

and are produced by, respectively:

`\ds \sum_{i \mathop = 1}^n`

`\ds \prod_{i \mathop = 1}^n`

`\ds \frac {-b \pm \sqrt {b^2 - 4ac} } {2 a}`

`\ds \lim_{n \mathop \to \infty} \frac 1 n`

Furthermore, to improve aesthetic appeal certain characters, such as $=$ and $\in$, when used in subscripts of big operators, must be endowed with the `\mathop`

command to enforce appropriate spacing.

As a contrast, compare:

- $\ds \sum_{i = 1}^n \quad \sum_{i \mathop = 1}^n$
- $\ds \bigcap_{n \in \N} \quad \bigcap_{n \mathop \in \N}$

The `\mathop`

command is to be used in the following manner (the code produces $\ds \sum_{i \mathop = 1}^n$):

\ds \sum_{i \mathop = 1}^n

#### Abbreviated Symbols

Certain symbols have abbreviated forms for their big versions:

`\dfrac`

can be used instead of`\ds \frac`

`\dbinom`

can be used instead of`\ds \binom`

where `d`

is for `display`

.

Of course, if other big operators are used in the same equation, the `\ds`

command is needed anyway.

However, it does no harm to include `\dfrac`

and `\dbinom`

inside a line defined as `\ds`

, and may make refactoring easier.

So feel free to develop the habit of using `\dfrac`

and `\dbinom`

throughout.

### The d of Calculus

When writing calculus operators, use a non-italic form for the $\d$. To achieve this, write it as `\d`

or `\rd`

(the latter includes a half-space before it, for use in integrals).

So you would have:

- $\dfrac {\d y} {\d x}$

which would be produced by:

`\dfrac {\d y} {\d x}`

rather than:

- $\dfrac {d y} {d x}$

which would be produced by:

`\dfrac {d y} {d x}`

### Fonts

We have several fonts available, many of which have particular conventional uses in mathematics.

Examples are:

- Calligraphy:
`\mathcal`

, which produces $\mathcal{ABCDE} ..., \mathcal {1234567890}$ (uppercase only, but also digits) - Blackboard:
`\mathbb`

or (preferably)`\Bbb`

, which produces $\Bbb{ABCDE} ...$ (uppercase only, no digits) - Script:
`\mathscr`

, which produces $\mathscr{ABCDE} ...$ (uppercase only, no digits) - Sans serif:
`\mathsf`

, which produces $\mathsf{ABCDE} ... \mathsf{abcde} ..., \mathsf {1234567890}$ - Fraktur:
`\mathfrak`

, which produces $\mathfrak{ABCDE} ... \mathfrak{abcde} ..., \mathfrak {1234567890}$ - Fixed Width:
`\mathtt`

, which produces $\mathtt{ABCDE} ... \mathtt{abcde} ..., \mathtt {1234567890}$

The use of Fraktur and Script are discouraged, as they are not so easy on the eye and can be difficult to decipher on certain browsers.

Also note that:

`\T`

and`\F`

implement $\T$ and $\F$ respectively, designed to be used for True and False respectively- $\N, \Z, \Q, \R, \C$ have their own $\LaTeX$ codes:
`\N, \Z, \Q, \R, \C`

- We also have
`\GF`

implemented for $\GF$ (note that`\F`

cannot be used here, as it is already used for $\F$) - all
`\mathcal`

uppercase letters have custom $\mathsf{Pr} \infty \mathsf{fWiki}$ shortcut implementations, as`\AA`

,`\BB`

, and so on

### Use of Logical Symbols in Mathematical Exposition

This applies mainly to the use of the conjunction symbol $\land$, that is $\text {and}$, and the disjunction symbol $\lor$, that is $\text {or}$.

It is convenient sometimes to write a statement in the style:

- $\forall y \in R: \lambda_y = y * I_{_R} \land \rho_y = I_{_R} * y$

However, it may not be immediately obvious to the reader exactly what $\land$ means.

In the various fields, for example abstract algebra and set theory, $\land$ and $\lor$ have a number of different meanings, for example meet and join.

If the reader has been studying such material, it can be irritating to have to change mental gears and suddenly have to adjust to the fact that $\land$ means $\text {and}$.

Hence it is strongly recommended that the above statement be written:

- $\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$

reserving $\land$ and $\lor$ for their use in the field of logic.

### Punctuation niceties

A sentence broken by a displayed equation should be ended with a colon:

- $\dfrac {\text{display}} {\text{equation}}$

for a better presentation.

On the other hand, the displayed equation itself should not be ended with a full stop or comma.

That is, one should write:

- $\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S$

and not:

- $\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S$.

In particular, including the full stop *inside* the $\LaTeX$ it terminates is *definitely* incorrect, for readily apparent reasons. So please do not do this:

- $\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S.$

This is a style tip borrowed from Ian Stewart, from his *Galois Theory, 3rd ed.* of $2004$.

Use of commas is discouraged. This sort of structure is considered incorrect:

*Let $x$ be as follows,*

- $x \in S$

as commas are reserved in mathematics for separation of elements of a list.

### Q.E.D.

To end a proof, the template `{{qed}}`

should be used, which looks like:

$\blacksquare$

or if you wish to break your page up into subproofs, end those subproofs with `{{qed|lemma}}`

, which looks like:

$\Box$

In a dash for consistent notation, it is understood that these templates should immediately succeed the last line of the proof, that is:

Hence the result. {{qed}}

and not:

Hence the result. {{qed}}

Tempting though it is to write "Q.E.D." at the bottom, this is so uncool as to be positively naff.

## $\LaTeX$ Code Style

There are a few general source code conventions which make your code easier to read and maintain:

- Each variable, and each command beginning with a backslash should be preceded by a space, except (for some unexplained result of evolution) when enclosing things in brackets. See some of the above instance for a typical example.

- When enclosing brackets around an object, always use the
`\paren { ... }`

command, for example`\paren {a + b}`

rather than`(a + b)`

.

- There should be no need to use the commands
`\big, \Big, \Bigg`

etc. for specifying the sizes of parentheses. Using the`\paren {...}`

technique (as above) will almost always automatically size the brackets aesthetically.

- Punctuation should appear (if it is really necessary) outside the $\LaTeX$ environment, for example:

- Hence $\map f {\sin x}$. (produced by:
`$\map f {\sin x}$.`

)

- Hence $\map f {\sin x}$. (produced by:

- as opposed to:

- Hence $\map f {\sin x}.$ (produced by:
`$\map f {\sin x}.$`

)

- Hence $\map f {\sin x}.$ (produced by:

- Single-character parameters to standard $\LaTeX$ constructs need not be put in curly braces. That is,
`\dfrac 1 2`

is preferred to`\dfrac {1} {2}`

. They both produce: $\dfrac 1 2$

It makes the source code significantly easier to read.

Having said that, please do not ignore the rule about spacing. The same effect can also be achieved by `\dfrac12`

(see, it still looks like $\dfrac12$) but that is significantly harder to interpret visually.

### Aligned Material

If an equation includes multiple equalities or inequalities, it is best to place each equality on a new line.

For example:

- $\dfrac \d {\d t} \map H U = \d \map H U \cdot \dot U = \map \Omega {\map {X_H} U, \dot U} = \map \Omega {\map {X_H} U, \map {X_H} U} = 0$

would look better as an aligned equation. This is done using the following commands:

{{begin-eqn}} {{eqn | l = | r = }} ... {{eqn | l = | r = }} {{end-eqn}}

Here, the section following `| l =`

is a $\LaTeX$ environment, and should contain anything you want to appear to the left of the equals sign.

The section following `| r =`

is the same, but will appear to the right of the equals sign.

`| ll =`

and `| rr =`

are similar, but produce material in columns further to the left and further to the right respectively. In particular, the `ll`

column is often used for an "implies" or "leads to" sign where the `l`

and `r`

are used for either side of a string of equations.

All these $\LaTeX$ environments are **already** in `\displaystyle`

mode, so there is no need to include that command in your equation.

When entering such an `{{eqn}}`

environment, it should globally look like this:

{{eqn | l = 1 + 1 | r = 2 }}

That is, it adheres to the following principles:

- Every empty column should in general be omitted, except perhaps for
`| c =`

sections, which can be left as placeholders for possible future addition of comments - Non-empty columns are entered on separate lines, with the
`|`

and`=`

all aligned.

These conventions serve to optimize readability.

More options and abilities of the `{{eqn}}`

can be found on its page, `{{eqn}}`

.

The section following `| c =`

is *not* a $\LaTeX$ environment, and can be used to add any comments about the equation at this point.

So the example we gave above would be typeset as:

{{begin-eqn}} {{eqn | l = \frac \d {\d t} \map H U | r = \d \map H U \cdot \dot U | c = [[Chain Rule for Derivatives]] }} {{eqn | r = \map \Omega {\map {X_H} U, \dot U} | c = Definition of $X_H$ }} {{eqn | r = \map \Omega {\map {X_H} U, \map {X_H} U} | c = [[Hamilton's Equations]] }} {{eqn | r = 0 | c = from above: $\Omega$ is [[Definition:Skew-Symmetry|skew-symmetric]] }} {{end-eqn}}

The finished result will look like:

\(\ds \frac \d {\d t} \map H U\) | \(=\) | \(\ds \d \map H U \cdot \dot U\) | Chain Rule for Derivatives | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \Omega {\map {X_H} U, \dot U}\) | Definition of $X_H$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \Omega {\map {X_H} U, \map {X_H} U}\) | Hamilton's Equations | |||||||||||

\(\ds \) | \(=\) | \(\ds 0\) | from above: $\Omega$ is skew-symmetric |

The operator that is displayed in this template can be changed using `| o = `

to show inequalities, etc.

Note the following:

- Do not include two consecutive open or close curly braces:
`{{`

or`}}`

**anywhere**in your`{{eqn}}`

templates. It will break the interpreter.

Put spaces in: `{ {`

or `} }`

and it will be okay.

- Do not include the vertical line
`|`

(a.k.a. "pipe") in $\LaTeX$ expressions as this also breaks the interpreter. Use`\vert`

(or`\lvert`

and`\rvert`

) instead.

- In particular,
`\|`

(used to produce $\|$) has the same problem. Use`\Vert`

etc. instead.

These caveats apply only within the `{{eqn}}`

environment. Elsewhere on the page such constructs should be fine. To accommodate for the inevitable copy-paste efforts, and for consistency's sake, it is however desirable to always use `\vert`

and `\Vert`

, and to insert a space between adjacent curly braces within $\LaTeX$ strings.

## Linguistic Style

### Language

This is an English language website, and so all pages are to be presented in English.^{[1]} Where there is a difference between spellings between US and rest-of-world English, the US version is generally used, with a few exceptions (the spelling of metre is under discussion).

### Linguistic Style

During the presentation of a mathematical argument, a formal style is preferred.

For example:

*Suppose that ...*

is preferred to:

*Let's suppose that ...*

and:

*Hence the result.*

is preferred to:

*... and we're done.*

As an attempt is being made for $\mathsf{Pr} \infty \mathsf{fWiki}$ to appeal to as wide an audience as possible worldwide, using colloquial language (except for example when illustrating logical concepts by means of everyday examples) is discouraged.

### "Let" and "Suppose"

It is preferred that "Let" is used to introduce the existence of an entity in an argument, as follows:

- Let $S$ be a set.

- Let $x, y \in S: x \ne y$.

- $\ldots$

However, when introducing an entity whose existence is in question (for example, when constructing a Proof by Contradiction), the word "Suppose" is recommended:

- Suppose $T \subseteq S$ such that $\card T > \card S$.

- $\ldots$

### "Any"

The word "any" can be ambiguous.

It is recommended that it not be used.

Instead, consider whether "every" or "an arbitrary" can be used instead.

### Abbreviations

The difference between "e.g." (*exempli gratia* - for example) and "i.e." (*id est* - that is) is sadly falling into obscurity. It is all too common for "i.e." to be used when "for example" is meant, and vice versa.

So as to remove all confusion, such abbreviations are discouraged.

Also, beware the ubiquitous confusion between *its* and *it's*. The full version *it is* should be used instead of *it's* in any case, so *it's* should have no reason to appear.

### Sentence Length

During the course of an argument to present a mathematical proof, follow these rules:

- Each sentence should be short.

- Each sentence should convey one step, either:
- One simple statement, or:
- One compound statement of the form:
*$P$, therefore $Q$*.

- Each sentence should be on a separate line.

Compare the presentations:

$(1):$

*$S$, because of $R$ (we know this from Tom's Theorem), because of $Q$ (from above) which applies when $P$ holds (see Fred's Theorem), but we know $P$ holds because it's what we defined in the first place.*

$(2):$

- Let $P$ hold.
- From Fred's Theorem, it follows that $Q$.
- From above, $R$.
- From Tom's Theorem, $S$.

The following is an example of the style of mathematical exposition which we believe has no place in $\mathsf{Pr} \infty \mathsf{fWiki}$, and indeed, the entire universe:

*The ($\implies$) is shown just the same as above, while the other direction easily follows, since $\MM$ satisfying the condition that for every $\LL$-formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$, if there is an $n$ in $\NN$ such that $\NN \models \map \phi {n, \bar a}$, then there is an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$, is closed under functions (by directly applying the condition to formulae of the form $\map \phi {x, \bar y} = \paren {x = \map f {\bar y} }$), and hence the universe of a substructure, which reduces it to the statement above.*

Here's an even worse example, posted up by an editor whose approach to contribution is so contrary to house style that appears to be deliberate trolling:

*If 24*k with k coprime to 6 has exactly 120 divisors, than k has exactly 15 divisors, thus k is a square number, thus k cannot be == 5, 7, 11 mod 12 (since 5, 7, 11 are not quadratic residues mod 12), thus a number == 120, 168, 264 mod 288 cannot have exactly 120 divisors (since such numbers can be written as 24*k with k coprime to 6 and k == 5, 7, 11 mod 12), thus if there are 120 consecutive integers with exactly 120 divisors, than the start number must be == 0, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287 mod 288, and hence == 0, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 mod 32, thus there are 4 consecutive multiples of 32 among these 120 integers, and one of these 4 numbers must be == 64 mod 128, thus the number of divisors of this number must be divisible by 7 and cannot be 120, which is a contradiction!*

Please, don't do this. Just don't.

#### Parenthetical Notes

It is often tempting to slide some further information or explanation into the middle of a sentence by slipping it into parenthesis.

It is **highly recommended** that this practice be avoided as far as possible.

Not only does this clash with the $\mathsf{Pr} \infty \mathsf{fWiki}$ house style of short, simple sentences, but also can cause confusion when the parenthesized material contains similar mathematical notation:

- Let $\map {f_n} x \in \Sigma$ (where $\map {f_n} x$ is a family (indexed by $\map I \alpha$) of functions in $\struct {\Sigma, \mu}$) be a function such that ...

So please don't do this.

A similar stylistic presentation to be avoided is the following

- The supremum of $f$, $\map \sup f$, is defined as:
- $\map \sup f = \map \max {\Img f}$

It is clear from the line following that $\map \sup f$ is how the supremum of $f$ is denoted, so there is no need to include it as a parenthetical explanation in the first line.

This is preferred:

- The supremum of $f$ is defined as:
- $\map \sup f = \map \max {\Img f}$

If you are uneasy about the ability of the reader to make that connection, feel free to write something like:

- The supremum of $f$ is defined and denoted as:
- $\map \sup f = \map \max {\Img f}$

### Filler Words

Whether or not filler words are needed (*it follows that*, *we have*, *hence* etc.) is a stylistic decision. Fewer words are preferred, but clarity and completeness override every other consideration.

The general approach is to try to use as terse a form as possible.

Compare:

- We have that the ordinal subset of an ordinal is an initial segment of it, so it follows that:

with:

The latter form is preferred.

#### Definitions

In definitions, in particular, it is often tempting to fill up the internet with linguistic constructs like:

- $X$ is called a
**some object**

- $X$ is known as
**some epithet**

- $X$ is said to be
**some property**

- $X$ is described as being
**some type**

The following are preferred:

- $X$ is
**some object**

and so on.

Just use **is**.

### Empty Statements and Waffle

It is tempting to fill a page up with statements that do not actually impart any information, but which make the author look and feel good.

Such are to be avoided.

Examples:

*The first part of the proof is easy.*

*We mention for the interested reader ...*

*This is trivial:*

See also the templates `{{handwaving}}`

and `{{explain}}`

.

This article is complete as far as it goes, but it could do with expansion.In particular: There are plenty more -- these will be added as they are encountered.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Capital Letters begin Sentences

This is raised as a particular point, because it crops up over and over again.

The sentence form in question is:

- Let (such-and-such) hold, where (so-and-so) means (thus and so).

When (such-and-such) is a statement in mathematical symbols, placed on its own line (as per house style recommendations), the temptation is to present the above sentence as:

- Let:
- $\ds S = \sum_{i \mathop \in \N} \frac 1 {2^i}$

- Where $\ds \sum$ denotes summation.

Just because it starts a new line does *not* mean that "where" is to be written with a capital W. It is the continuation of the previous sentence, which just happens to have, as part of its main clause, a mathematical expression.

It should be:

- Let:
- $\ds S = \sum_{i \mathop \in \N} \frac 1 {2^i}$

- where $\ds \sum$ denotes summation.

Breaking this linguistic rule can lead to confusion, especially when the "where" clause starts to get complicated:

- Let:
- $\ds S = \sum_{j \mathop \in \N} \lim_{x \mathop \to \infty} \cos j x + i \sin j x$

- Where $\ds \sum$ denotes summation and $\lim$ is the limit as $x$ tends to infinity and:
- $\cos j x + i \sin j x = e^{ijx}$

In the above, the reader, thinking that "where" starts the next sentence, and therefore a new thought, is left wondering:

- "Where this applies, and that means that, and this ... then what?"

whereas in fact the only reason for the "where" clause is to amplify the sense of the expression above it.

Similarly:

\(\ds x \in A \cap \paren {B \cap C}\) | \(\leadstoandfrom\) | \(\ds x \in A \land \paren {x \in B \land x \in C}\) | By definition of Set Intersection | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in A \land x \in B} \land x \in C\) | Rule of Association: Conjunction | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {A \cap B} \cap C\) | By definition of Set Intersection |

In the above, the "by definition" phrases in the comment column should not start with a capital letter, as they continue the "sentence" started on the left.

Thus the above structure is better rendered as:

\(\ds x \in A \cap \paren {B \cap C}\) | \(\leadstoandfrom\) | \(\ds x \in A \land \paren {x \in B \land x \in C}\) | by definition of Set Intersection | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in A \land x \in B} \land x \in C\) | Rule of Association: Conjunction | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {A \cap B} \cap C\) | by definition of Set Intersection |

Better still, lose the redundant filler-word "by", and render the entire structure elegantly as:

\(\ds x \in A \cap \paren {B \cap C}\) | \(\leadstoandfrom\) | \(\ds x \in A \land \left({x \in B \land x \in C}\right)\) | Definition of Set Intersection | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in A \land x \in B} \land x \in C\) | Rule of Association: Conjunction | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {A \cap B} \cap C\) | Definition of Set Intersection |

Now, as there is no filler-word "by", the comment is no longer implicitly part of a sentence, and so the comment is a standalone label which now merits an uppercase presentation.

Here, note that a further evolutionary step has been made: to replace the code `Definition of [[Definition:Set Intersection|Set Intersection]]`

with the template construct `{{Defof|Set Intersection}}`

for further streamlining of the source.

- Beware
- This
`{{Defof}}`

template was designed specifically for the`c`

parameter of the`{{Eqn}}`

template.

It is *not* for using in the body of an exposition, specifically because of the fact that it has been designed to start with a **capital letter**.

## Sources

It is good to indicate where the information comes from. This is done in $\mathsf{Pr} \infty \mathsf{fWiki}$ in the last of the page in a section called Sources.

## Adding sources

If there are multiple sources, they are to be listed first in **chronological order**, then alphabetically on the name of the (first) author.

As stated on Help:Page Editing, the sources should be using a bulleted list, ordered by date of publication of the edition cited, and after that alphabetically, sorted on the surname of the (first) author.

For example (an excerpt of the Sources section of Definition:Set Union):

== Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Union of Singleton|next = Union with Empty Set}}: $\S 4$: Unions and Intersections * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Equality of Sets|next = Definition:Set Intersection}}: $\S 1.1$: Definition $1.2$ * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Definition:Set Union/General Definition|next = Union is Commutative}}: Introduction to Set Theory: $1$. Elementary Operations on Sets * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Empty Set Subset of All|next = Intersection Subset Union}}: $\S 1.4$ * {{BookReference|Modern Algebra|1965|Seth Warner|prev = Associative and Anticommutative|next = Definition:Set Intersection}}: $\S 3$

## Types of sources

There are several templates that can be used:

### Hardcopy Sources

- Template:BookReference
- This is used to reference a specific book which will have been documented in the Books page. The idea of this is that if you have sourced the information for a page directly from a book, then it should be possible to provide the details of that book.

Example:

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*: $\S 4.17$: Theorem $28$

which can be found on the page Characteristic times Ring Element is Ring Zero.

- Template:Citation
- This is used to reference a specific article in a journal. This is still under development, as the individual Journal entries still need to be worked on.

Examples of their use can be found on various Mathematicians pages, for example:

- 1908:
*Mathematical Logic as Based on the Theory of Types*(*Amer. J. Math.***Vol. 30**: pp. 222 – 262)

which appears on the page for Bertrand Russell.

The style of this is still evolving.

### Online Sources

There are templates for the following online sources. Each one has been crafted so as to produce a reference in the style requested by the online source in question.

- Template:MathWorld
- This provides a direct link to a page on the https://mathworld.wolfram.com/ website.

Example:

- Weisstein, Eric W. "Circular Sector." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/CircularSector.html

which can be found in the page Area of Sector.

- Template:Planetmath
- This provides a direct link to a page on the https://planetmath.org/ website.

Example:

*This article incorporates material from Urysohn's Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

which can be found in the page Urysohn's Lemma.

- Template:MacTutor Biography
- This provides a direct link to a page on the https://mathshistory.st-andrews.ac.uk/ website.

Example:

which can be found in the page for Hanna Neumann. Note that the link presentation is taken from the page the template is invoked from.

- Template:KhanAcademy
- This provides a direct link to the Khan Academy.

Example:

- For a video presentation of the contents of this page, visit the Khan Academy.

which can be found in the page Limit of Sine of X over X at Zero/Geometric Proof.

- Template:Metamath
- This provides a direct link to Metamath.

Example:

which can be found in the page First Principle of Transfinite Recursion.

- Template:Mizar
- This provides a direct link to Mizar.

Example:

- Mizar article TOPGEN_1:10

which can be found in the page Characterization of Boundary by Basis.

- Template:OEIS
- This provides a direct link to the On-Line Encyclopedia of Integer Sequences.

Example:

- This sequence is A002193 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

which can be found in the page Square Root of 2 is Irrational.

- Template:SpringerOnline
- This provides a direct link to a page on the Springer Online Encyclopedia of Mathematics.

Example:

- Ring. O.A. Ivanova (originator),
*Encyclopedia of Mathematics*. URL: https://www.encyclopediaofmath.org/index.php?title=Ring

which can be found in the page Definition:Ring (Abstract Algebra).

- Template:Stackexchange
- This provides a direct link to a page on Mathematics Stack Exchange.

Example:

- robjohn (https://math.stackexchange.com/users/13854/robjohn), How to prove that $\ds \lim \limits_{x \mathop \to 0} \frac {\sin x} x = 1$?, URL (version: 2013-06-19): https://math.stackexchange.com/q/75151

which can be found in the page Derivative of Sine Function/Proof 5.

### Acceptability of Online Sources

**NOTE: The above are currently the ONLY web resources which are to be used as general citation sources.**

Others may be added to the above as and when they come to our attention as being particularly useful.

So feel free to challenge this assertion if you find something which appears to be a particularly rich and productive resource.

Scholarly papers which are available online may usually also be cited.

What are *not* generally acceptable include:

- Lecture notes for university courses available online (because they do not stay online forever, and this causes dead links)
- Links to pages in homework help forums
- Discussion pages in
*any*web forum - Wikipedia -- not because we don't like them, but because as they are self-proclaimed tertiary source, there is no need to do so -- we would rather go to the actual source works. See also Wikipedia:Citing Wikipedia.

## Splitting sources

In some cases it is necessary to split a referenced theorem, proof or definition into multiple pages, because for example:

- a theorem contains multiple statements
- a proof contains in fact multiple proofs
- a definition defines multiple concepts at once.

If so, the source has to be referenced at every page, and its process flow is updated according to the order in which the elements appear in the source.

## References

- ↑ Suggestions have been made as to how we may go about the exercise of internationalization, but progress in that direction is slow due to its perceived low priority.