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I know that if you want to refer to the set $ A $ with the element $ x $ added, you can write $ A \cup \{x\} $. But is there a common shorthand for this?

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I’m not aware of one. – Brian M. Scott Jan 6 '13 at 20:07
In very informal writing such as notes to yourself, you might write $A + x$ instead, but otherwise $A \cup \{x\}$ is about the best there is. – Ted Jan 6 '13 at 20:08
I have never seen one. – ncmathsadist Jan 6 '13 at 20:11
Such a notation would save at best two characters anyway... – Najib Idrissi Jan 6 '13 at 20:26
@nik: This is not always about the amount of characters one has to type. It can be about readability and clarity of expressions. I used to think so as well, but when I wrote my thesis and had more than a few places were such "one element addition" was needed, it was quickly apparent that a global notation is needed, and it turned out useful in another context as well. – Asaf Karagila Jan 7 '13 at 1:16

There is no particular notation that I am aware of.

If you have a particular set in mind you can always write something such as:

We shall write $A(x)$ for the set $A\cup\{x\}$.

This is just a suggested notation, of course. Be careful that the readers won't confuse this with a function symbol (although it is a function symbol if you think about it). It might be easier to use $A_x$ in some cases (if font sizes are not bothering).

Whatever you do, though, write the explicit notation in your text.

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"$\mathbb N$ denotes the set of natural numbers, $\emptyset(0)(1)(2)(3)\cdots$" :) – Rahul Jan 6 '13 at 20:52
@Jasper: Some people will disagree with this title, surprisingly enough - I am one of them. – Asaf Karagila Jan 6 '13 at 21:00
@Rahul: But formulas are finite, and your notation is not. :-) – Asaf Karagila Jan 6 '13 at 21:01
@Rahul: You could even write $\mathbb{N} = \{\emptyset,\emptyset(\emptyset),\emptyset(\emptyset) (\emptyset(\emptyset)), \emptyset(\emptyset) (\emptyset(\emptyset))(\emptyset(\emptyset) (\emptyset(\emptyset))),\emptyset(\emptyset) (\emptyset(\emptyset))(\emptyset(\emptyset) (\emptyset(\emptyset)))(\emptyset(\emptyset) (\emptyset(\emptyset))(\emptyset(\emptyset) (\emptyset(\emptyset)))),\ldots\}$ – Trevor Wilson Jan 6 '13 at 22:29

Logicians do have a convention of writing the likes of $\Gamma, A \vdash (A \lor B)$ when officially -- since $\Gamma$ [by convention] is a set of premisses, and $A$ is an additional premiss, and the derivability relation relates a set of wffs to a wff -- they mean $\Gamma \cup \{ A\}\vdash (A \lor B)$. This shorthand convention obviously avoids some clutter.

This usage -- where similarly, $\Gamma, A, B$ means $\Gamma \cup \{A\} \cup \{B\}$ -- although very common, seems to local to logicians, and perhaps only(?) used when talking of sets of wffs. I can't remember noticing it being used in other contexts where set notation is used.

But I suppose if it did save enough repeated clutter to be worthwhile, you could borrow the logicians' convention and write $A, x$ (especially if symbols are clearly typed, as in the logicians' usage, so it is plain which indicate sets of a certain kind and which their elements).

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I've seen books use $A;x$ to define $A\cup\{x\}$, although they always make sure to define it before hand.

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Hi! Can you please reference any of such books? – MattAllegro Sep 6 '15 at 14:22
I have 'A Mathematical Introduction to Logic' by Herbert B. Enderton in mind, though I remember seeing it elsewhere before, I just can't remember where – user268417 Sep 6 '15 at 14:25

Sometimes, when adding a basepoint to a topological space $X$, one writes $X^+$ for the resulting topological space. It is built on the set $X\cup\lbrace *\rbrace$. You could hijack this notation for your own personal use.

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