Is there any notation that can represent any member of a particular set? For example, how could I say that I want any real number, or any imaginary number?
I do know there is a symbol to denote "for all", $\forall \Re$, but I don't want that. I want to express that for example that a operation will return any member of the real set.
Is there such a notation?
This concept has been used extensively before, I believe it was introduced by Hilbert as the ε-operator in a logical calculus called the ε-calculus. The syntax is $ε\ x\ A(x)$ or $ε\ x.A(x)$ which mean any $x$ satisfying the property $A$. In your case the property you are using is the membership in a set, so you would write $ε\ x.x \in A$ (though I'm not sure if this was allowed in the original calculus). The ε-operator is also used in some formal proof systems, such as HOL (Higher Order Logic).
See also the entry in the Stanford Encyclopedia of Philosophy.