How to make the distinction between $a$ is some element in $\mathbb R$ and $a$ can be any element in $\mathbb R$

Question

This is a question relating to the notation. I think I have often seen the $a \in \mathbb R$ refer to that $a$ can be any element in $\mathbb R$. But of course, this is not necessary so, for example $a$ could be restricted to numbers 1 and 2, and still be in $\mathbb R$, in which case $a \in \mathbb R$ is also used and is more strictly correct.

So my question is, how can we specify that $a$ is restricted to some elements that belong in $\mathbb R$ and conversely that $a$ can be any element in $\mathbb R$? What notation should be used to differentiate the two cases?

In linear algebra there are some cases where it's desirable that $a$ can be any element in the real numbers, rather than just belonging into the set. That's the motivation for the question.

Answer 1

$a \in \mathbb{R}$ means that $a$ can be any element of $\mathbb{R}$. If you want to restrict $a$ to be in some specific subset A of $\mathbb{R},$ you would say $a \in A \subset \mathbb{R}.$

Answer 2

I think that in general, you should use the "smallest" set that is applicable. And you know that $\mathbb R$ is uncountably infinite. $\mathbb Z$ is also infinite, but countably so. And $\{1, 2\}$ is not just finite, it is trivially countable. It's clearly the smallest of these sets

So if $a$ can really only be one of 1 or 2, then you should really write $a \in \{1, 2\}$. Sure, we can try to come up with exotic scenarios in which it would not necessarily follow that $a$ is a real number, but for practical purposes, it's almost always unnecessary.

Something that comes up more often is when $a$ can be any real number except 0, though it can be a number that is arbitrarily close to 0, e.g., $10^{-(10^{10})}$. You could use $\mathbb R^*$, but this is not universally accepted. Or you could just use words (an option so often overlooked) and say "$a$ is a nonzero real number."

Something else that you might actually encounter is that $a$ can be any real number between 1 and 2, like $\sqrt 2$ or $2 - 10^{-(10^{10})}$. Then you could write something like $1 \leq a \leq 2$. Or you could use interval notation.