Is $\mathbb{Z} \times \mathbb{R} \subseteq \mathbb{R} \times \mathbb{Z}$ true?

Question

So with the question being true or not:

$\mathbb{Z} \times \mathbb{R} \subseteq \mathbb{R} \times \mathbb{Z}$

I found that it is not true because $\mathbb{R}$ is not a subset of $\mathbb{Z}$ for the $y$ coordinates. Is this true? Thanks.

Answer 1

Yes, it is not true. It is not a subset strictly speaking. But both sets are isomorphic in a very obvious way, so you can actually identify both sets (by permuting the coordinates) and then write a subset by abuse of notation (as $A\subseteq A$).

Answer 2

It is true, because the statement $$\forall a\in\mathbb Z\times\mathbb R: a\in\mathbb R\times\mathbb Z$$ (which is the definition of $\mathbb{Z} \times \mathbb{R} \subseteq \mathbb{R} \times \mathbb{Z}$) is not true.

The statement can be proven to not be true because we can see that $$\exists a\in\mathbb Z\times\mathbb R:a\notin\mathbb R\times\mathbb Z $$ (the negation of the above statement) is true, and it can be proven by taking $a=(0,\frac12)$.