Is there a shorthand notation for adding an element to a set? 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? 
 A: 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.
A: I've seen books use $A;x$ to define $A\cup\{x\}$, although they always make sure to define it before hand.
A: 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).
A: I know it is an old question, but I was searching for something related, and I did not find anything and I would like to share here what I came up with - this is actually not a shorthand, but the opposite, I think more things are to be denoted with this operation.
If we have a set $A$, and we add an element, our understanding is that we have the same set just more "filled", but can we say that it is the same set? I mean, we might still denote it as $A$, but it has at least one problem, as it was mentioned in a comment also, that in math we should not write this as we would in programming languages: $A=A\cup\{a_i\}$. We might use $A^\prime$ to denote that this is a variant of $A$, and use $A^{\prime\prime}$, $A^{\prime\prime\prime}$, and so on for subsequent additions of elements, and then we can write $A^\prime=A\cup\{a_i\}$, $A^{\prime\prime}=A^\prime\cup\{a_j\}$, $A^{\prime\prime\prime}=A^{\prime\prime}\cup\{a_k\}$, and so on, but this notation is not suitable if we have aribtrary additions and we want to write down the general case. So I think we need a kind of notation for the state of the set, let's say $s\in\mathbb{N}$, which starts at $0$ as the initial state, so when we define our set the first time we denote it as $A^0$, and then we can write the addition as
$$A^{s+1}=A^s\cup\{a_i\}$$
This way we can refer to the set before/after certain elements are added, and we can also calculate the number of elements based on the state value.
A: 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.
