Why start at x, y, z - User contributions [en-gb]

Set notation

2021-07-12T16:47:27Z

Hpecora1: Fixed TeX in paragraph I added earlier.

[[Category:Local variations]]
[[Category:Ambiguities]]

When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside:

* All the elements of the set, separated by commas: $ \{a,b,c, \ldots\} $
* Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: $ \{ x \mid P(x) \} $ or $ \{ x : P(x) \} $.

Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the $a_i$.

Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:

$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$

Combinations

2021-07-12T16:45:36Z

Hpecora1:

[[Category:Local variations]]
[[Category:Needs filling in]]

Is it $_nC_r$ or $^nC_r$?<ref>[https://twitter.com/christianp/status/335024096947617793 Tweet by Christian Lawson-Perfect]</ref>

More commonly used is the binomial coefficient notation $ \binom{n}{r}$.

Combinations

2021-07-12T16:43:32Z

Hpecora1:

Circumflex to Distinguish Variable Names

2021-07-12T16:40:42Z

Hpecora1: Added common readings of diacritical marks.

[[Category:Unpleasantness]]
[[Category:Variable Names]]

Some authors create new variable names by adorning well-known symbols (like $ A $) with the upright ($ \hat{A} $), read “A hat”, and inverted ($ \check{A} $) circumflex, read as “A check”. Depending on the font, these tiny diacritical marks can be hard to discern at a glance. A particularly egregious example is found in Hinze 2012, [http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf "Generic Programming with Adjunctions"] (p. 5).

[[File:Hinze2012 adjunctions circumflex.png|frame|center|Hinze 2012, [http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf "Generic Programming with Adjunctions"] (p. 5) creates two new variable names by adorning the letter A with upright and inverted circumflex symbols.]]

Stacked fractions

2021-07-12T16:38:39Z

Hpecora1: Added reference for convoluted example due to Barry Mazur who was trying to annoy Serge Lang.

[[Category:Ambiguities]]
[[Category:Handwriting]]

A fraction written on multiple levels is often ambiguous, especially when handwritten. For example, $\frac{10}{\frac{2}{5}}$ and $\frac{\frac{10}{2}}{5}$ result in 25 and 1, respectively.

Things get even worse when you use the letter $\Xi$ in this convoluted example<ref>[http://www.ams.org/notices/200605/fea-lang.pdf AMS Notices: Remembrances of Serge Lang]</ref>:

[[File:Xi bar over Xi .jpg|thumb|center|alt=A handwritten stacked fraction with Xi bar divided by Xi. In effect it looks like nothing more than a stack of eight horizontal lines of varying sizes.|$\frac{\bar \Xi}{\Xi}$, handwritten]]

Set notation

2021-07-12T16:30:21Z

Hpecora1:

[[Category:Local variations]]
[[Category:Ambiguities]]

When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside:

* All the elements of the set, separated by commas: $ \{a,b,c, \ldots\} $
* Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: $ \{ x \mid P(x) \} $ or $ \{ x : P(x) \} $.

Piper H asks<ref>[https://twitter.com/pwr2dppl/status/1411816873826697224 tweet by Piper H]</ref>, in the notation $ \{ a_1, a_2, \ldots \} $, where the elements inside are a sequence, are you making an implicit claim that the elements are countable?

The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the \a_i.

Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg $f:\mathbb{R}\to\mathbb{R}$. This can cause ambiguities when defining a set containing functions, for example:

$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$