TITLE: In proofs, are "for each" and "for any" synonyms?
QUESTION [26 upvotes]: In proofs, are "for each" and "for any" synonyms?  Or some context is usually required to determine this?

REPLY [2 votes]: I would recommend avoiding "for any" because it can mean either "for some" or "for all", depending on context. That makes it easy to misunderstand, because you and your reader might disagree about which it does mean.
For example, compare:

Any person can downvote this answer;
If any person downvotes this answer, I will be sad.

The first of these is a universal statement: for all $x$, if $x$ is a person, $x$ can downvote. The second is existential: if there exists a person who downvotes, ... (If all people downvote then, well, I guess we get to discover if there's a negative rep cap, too!)
Those two examples are unambiguous but suppose somebody writes "A set $S$ is cromulent if, for any $X\subseteq S$, $f(X)=0$." Is that supposed to mean "for all $X\subseteq S$" or "for some $X\subseteq S$"?