Symbol denoting an element that is not in a set?

Question

When solving mathematical tasks it is often convenient to say something like:

Let $\#$ be an arbitrary element that is not present in set $X$

or

Let $\$$ be an arbitrary letter outside of alphabet $\mathcal{A}$

etc, etc. (Often because we will next construct a set / alphabet / algebra / whatever that is similar to the 'previous' one, but contains one more 'special' element).

I now have a problem because both $\#$ and $\$$ are used. Which other symbols are commonly used for this purpose?

I thought about $\bot$, but having read Wikipedia entry about this symbol I suppose I shouldn't use it unless I can show how my 'arbitrary element' translates to the notion of absurdum.

Besides indexing - can I ask for a list of symbols I may safely use for this purpose?

This answer proposes $\infty$ - but I must say this confuses me, because I've always associated this symbol to the notion of infinity.

Or would it be acceptable to invent my own symbols?

Let be an arbitrary element not included by the carrier of algebra $A$

_{I understand that the problem is rather minor and the points for an exam will likely not depend on this - but if I can improve, however slightly, the quality of my homework by not violating conventions and using symbols within their accepted meanings, then why not?}

Answer 1

Let $e$ be an arbitrary element that is not present in set $X$.

Let $l$ be an arbitrary letter outside of alphabet $\mathcal{A}$.

If all letters are already reserved (?):

Let $e^*$ be an arbitrary element that is not present in set $X$.

Let $l^*$ be an arbitrary letter outside of alphabet $\mathcal{A}$.

Answer 2

"Let $☺️ \not\in X$."? Works right up until your sets implement Unicode.

Upon consideration, although you haven't said as much, perhaps what you are really needing is a way to distinguish symbols in a language and symbols in a theory about that language. (This is a fairly common problem in formal language theory.) One method is to use the typographical convention you are already using. "$1$", "$2$", et c. are symbols in the language of arithmetic and they are not italicized. In fact, in proper style, we should never italicize constants, including $\pi$ and $\mathrm{e}$. (And, yes, I know that "$\pi$" is italicized and yes, I realize it is far too late to make Computer Modern Roman default to non-italic pi so that all of its typographical descendants can instead do the correct thing here.) So I recommend the same thing here. The set of your alphabet can be $X = \left\{ \mathrm{a}, \mathrm{b}, \mathrm{c}, \dots, \mathrm{z}, \mathrm{A}, \mathrm{B}, \dots, \mathrm{Z}, \dots \right\}$ and then continue using italics for symbols in the meta-language. E.g., $x \not \in X$.