Is there a shorthand notation for adding an element to a set?

Solution 1:

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.

Solution 2:

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

Solution 3:

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).