# How should I write the interval in conjunctionist notation?

I have this statement:

If $$\mathbb{A} = (-\infty, \infty)$$ write the interval in conjunctionist notation.

First I thought:: $$\forall{x} \in \mathbb{Z}:x \in \mathbb{A}$$

However, I am deleting irrational numbers, so after I did this:

Second answer: $$\forall{x} \in \mathbb{R}:x \in \mathbb{A}$$

However, I am deleting complex numbers, so finally is this:

$$\forall{x} \in \mathbb{C}:x \in \mathbb{A}$$, since the set of real numbers is a subset of the set of complex numbers.

But according to the guide my answer is incorrect and must be the second answer, and I have no idea why. Thanks in advance.

• $(-\infty, \infty)$ generally means all real numbers, not all complex numbers. – wgrenard May 29 at 0:51

## 1 Answer

The complex numbers, while orderable (since in bijection with the reals, which are canonically ordered), are not orderable in a way that is compatible with the operations there. Thus, since $$(-\infty,\infty)$$ indicates an interval, and thus implies some order, we must assume (unless given more information) that it indicates a subset of the reals--typically, the set of all real numbers.