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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.

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    $\begingroup$ I'm not clear what you are really asking here. If $a$ must be $1$ or $2$ it is still fine, if slightly odd, to say $a\in \mathbb{R}$. I suppose you could drive the point home by saying $a\in\{1,2\}\subset\mathbb{R}$ but this just seems silly. $\endgroup$ – The Count Apr 3 '17 at 23:01
  • $\begingroup$ Yeah I think generally if it is left as $a\in \mathbb R$ then this is to mean $a$ can take any value in $\mathbb R$. If $a$ takes only the values $1$ or $2$ we would say $a\in \{1,2\}$. Usually it is clear from the context, so we may say $\forall a \in \mathbb R$. If it is relevant that $a$ cannot take all values in $\mathbb R$ it should be stated, otherwise take it to be any $a\in \mathbb R$ $\endgroup$ – Aka_aka_aka_ak Apr 3 '17 at 23:06
  • $\begingroup$ As far as I am concerned, if no quantifier is explicitly stated, that means "for all." $\endgroup$ – Will M. Apr 3 '17 at 23:09
  • $\begingroup$ @TheCount if $a \in \{1,2\}$ and $a \in \mathbb R$ Then clearly $a$ can't be any element in $\mathbb R$. So I am asking what the distinction should be if we want to have a strict definition for these two different cases. $\endgroup$ – Dole Apr 3 '17 at 23:18
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    $\begingroup$ Well, then I think the two comments after mine took care of that. $\endgroup$ – The Count Apr 3 '17 at 23:19
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$a \in \mathbb{R}$ 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 $a \in A \subset \mathbb{R}.$

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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.

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  • $\begingroup$ Sorry to come back to this old question, but I think almost everyone misunderstood the question. It was about how to distinguish a case where $5\in \mathbb N$, so this is a statement "5 BELONGS to naturals", from $a \in \mathbb N$ "set a = any natural". I understand the same symbol is usually used for both of the different operations, which can be a problem. $\endgroup$ – Dole Jul 21 '17 at 10:04
  • $\begingroup$ There's no statute of limitations on mathematical questions. I'll try rereading your question later to see if I come up with anything new. $\endgroup$ – Robert Soupe Jul 22 '17 at 17:32

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