TITLE: Is there a shorthand notation for adding an element to a set?
QUESTION [21 upvotes]: 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?

REPLY [2 votes]: 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.