Does a variable always refer to just one of the possible elements in a given set? For example, if $x$ is an element of $S$, such that $x$ is a natural number greater than $5$, the possible values of $x$ are many, but does $x$ refer to only one of those possible values?
 A: In the statement, "Let $S = \{n\in\Bbb N~:~n\geq 5\}$ and let $x\in S$" then $x$ is an arbitrary single element of $S$.  An element may only be one value at any given time.  Although we do not know much information about $x$ other than the fact that it is an element of $S$, we are able to learn things about it, for example that $x! > 2^x$.  Such statements we can prove to be true regardless which exact value $x$ has.
If we were able to prove things about $x$ using only the knowledge that we are given and nothing more, then despite $x$ at the time only having been one value, since $x$ is arbitrary the argument works regardless what the value of $x$ was and so we can successfully prove things about every element of $S$ simultaneously with a single argument rather than multiple separate arguments.
A: Yes. It's something like the idea of common nouns in normal language use. Such nouns as man, country, plant, etc., refer to any one of a class of objects. Thus, they're like the variables of mathematics.
