How to write the union of sets

Question

This is just a question about notation(and I can not write it pretty well in Latex either).

Is $X=(0,+\infty)\subset\Bbb{R}$ and $Y=\Bbb{R}$.

Then $X\times Y= (0,+\infty)\times \Bbb{R} =$ ?

I've tried with:

1. $\bigcup_{x>0,x\in\Bbb{R},y\in\Bbb{R}}[x,y]$ (Sorry about the subindex)

2. $\{ \cup[x,y] : x,y\in\Bbb{R}, x>0 \}$ (I don't think it's correct)

3. $A =\{ [a,b] : a,b\in\Bbb{R}, a>0\}$ then $X\times Y=\cup_{[x,y]\in A}[x,y]$

4. Just $(0,+\infty)\times \Bbb{R}$ (but I need one more explicit)

How would you write it? Thanks!

Answer 1

Now, thank to @GEdgar, I can see all my options but 4 are wrong.

Just writing $(0,+\infty)\times \Bbb{R}$ or, using the definition of Cartesian Product, $\{ (x,y) : x,y\in\Bbb{R}, x>0 \}$ is correct.

Thanks.