# How to write the union of sets

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!

• Option 4 is correct, and most succinct. Or use the definition of $\times$ to write it out explicitly, $\{ (x,y) \colon x,y \in \mathbb R, x>0\}$. Commented Aug 24, 2014 at 3:15
• When intervals are involved, I prefer to use $\langle x,y\rangle$ for ordered pairs, and not $(x, y)$. Commented Aug 24, 2014 at 3:29
• @AsafKaragila, do you mean $\{\langle x,y \rangle : x,y\in\Bbb{R}, x>0\}$? Commented Aug 24, 2014 at 3:36
• Yes. That is what I mean. Because writing $(x, y)$ can be confused be the interval. Is $(1,3)$ an ordered pair or an open interval? Commented Aug 24, 2014 at 3:38
• It is an ordered pair. I was thinking in intervals, that's why I did that mess. I like your way, I think I didn't see it before in these cases. Commented Aug 24, 2014 at 3:44

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.