Is there notation for "for all unique…"? For instance, suppose you wanted to consider all distinct $x,y$ in some set $S$. Would we type $\forall !x,y\in S$?
Can we use "distinct" and "unique" interchangeably in this sense?
I've never seen notation for this and a google search/SE search did not find anything.
Best Answer
I'd be wary of using $\forall !$ as your notation for this. (See here for example.) The word 'unique' definitely doesn't mean what you want to say, though. 'Unique' means 'one', and 'distinct' means... exactly the opposite!
You'd be better off writing "for all distinct $x,y \in S$".
Working formally, if you were trying to show that a property $\phi(x,y)$ holds for distinct $x,y \in S$ you could write $$(\forall x,y \in S)(x \ne y \to \phi(x,y))$$ ...but outside of logic there isn't really much need for this kind of notation.