Writing Formal version of a statement

Question

How do i write the formal version of the following statement: No integer is both non-positive and non-negative unless it is the zero integer.

Should i split the statement into the form $p$ unless $q$ with the predicate domain as $\mathbb{Z}$?

I will appreciate a detailed explanation on the interpretation of the statement in its predicate statement form instead of only providing the answer. Thanks.

Answer 1

I always use the following trick when dealing with an 'unless': substitute 'if not'.

That is, 'P unless Q' becomes 'P if not Q'

Applied to your statement, we thus get: 'No integer is both non-positive and non-negative if it is not the zero integer'

Also, 'No P is Q' is the same as 'All P is not Q' or 'Any P is not Q', and thus we get:

'Any integers is not both non-positive and non-negative if it is not the zero integer'

Which translates to:

$$\forall x \in \mathbb{Z} (x \not =0 \rightarrow \neg (\neg x > 0 \land \neg x < 0))$$

Answer 2

$(\forall n\in\mathbb{Z}):n\geqslant0\wedge n\leqslant0\implies n=0$