Expressing the truth set of x≤0

Question

Say I have the following predicate which is in the domain of the integers (Z):

$P(x) : x \leqslant 0$

Would the truth set be expressed as:

$\{x \in\Bbb Z : x\leqslant 0\}$

or

$\{x \in\Bbb Z : x<0 \lor x=0\}$

or

$\{x \in\Bbb Z : x \in\Bbb Z^- \cup \{0\}\}$

Answer 1

Yes.   Those sets are equivalent and represent the set of integers which satisfy the given predicate.

Indeed, they can be simply represented as: $~~\Bbb Z^-{\cup}\{0\}$

Answer 2

Fact is that under $P(x) : x \leqslant 0$ and $\mathbb Z^-:=\{x\in\mathbb Z\mid x<0\}$ the following expressions are notations for exactly the same set:

• $\{x\in\mathbb Z\mid P(x)\}$
• $\{x \in\Bbb Z : x\leqslant 0\}$
• $\{x \in\Bbb Z : x<0 \lor x=0\}$
• $\{x \in\Bbb Z : x \in\Bbb Z^- \cup \{0\}\}$