What is the correct notation for "for all nonzero $a$ in the real numbers"?

Question

For all nonzero $a$ in the real numbers,...

Which, or neither, is the correct notation for the above quantification?

$\forall a \in \mathbb{R} \neq 0,\ldots$

$\forall a \neq 0 \in \mathbb{R},\ldots$

Answer 1

Neither would be particularly great.

The first seems to come with the additional implication that $\mathbb{R} \ne 0$, and the second does not necessarily imply that $a \in \mathbb{R}$, just that $0 \in \mathbb{R}$ and $a \ne 0$.

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For something like this, it would be best to write $$ \forall a \in \mathbb{R} \setminus \{0\} $$ since $\mathbb{R} \setminus \{0\}$ is the set of nonzero reals. I have a tendency to write $$ \forall a \in \mathbb{R}_{\ne 0} $$ since it is more compact, but this has a prerequisite of ensuring this is a well-defined notation (since it is somewhat infrequent in my experience). Another more common notation is $$ \forall a \in \mathbb{R}^\times \text{ or } \forall a \in \mathbb{R}^\ast $$ where $F^\times$ or $F^\ast$ can denote the invertible elements of a field (i.e. its nonzero elements).

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Or, frankly, you could just write "for all real, nonzero $a$" in plain English; no one would reasonably take issue with this outside of very narrow contexts, and it's perfectly clear. Math communication isn't just about writing down a mess of symbols, after all; to communicate effectively, plain English is useful too, especially if the thing you want to state mathematically is (frankly) clunky in symbolic language.

Answer 2

I would write either $\forall a \in \mathbb{R}\setminus\{0\}$ or integrate the $\neq 0$ part into the following, as in $\forall a \in \mathbb{R}: a\neq 0 \implies\ldots$.

Answer 3

I'd personally write it as $\forall a\in\mathbb{R}\setminus\{0\}$. Other forms might include "for all nonzero real $a$" and "$\forall a\in\mathbb{R}$ and $a\neq0$".