Are these partial order or total orders?

Question

I just want to know if the following are considered partially ordered sets or total set.

Here is the definition:

1. $\mathbb{R}^2$ : (a,b) R (c,d) iff $a\leqslant c$ and $b\leqslant d$

2. $\mathbb{R}^2$ : (a,b) R (c,d) iff $a\leqslant c$ or $b\leqslant d$

I believe that the number 1 is a partial ordered set and not a total set because it is not comparable. For example, (a,b) = (0,1) and (c,d) = (1,0) simply wouldn't work out.

Answer 1

The second isn't even an order, since an order has to be antisymmetric...i.e if $xRy$ and $yRx$, then $x=y$ for any order. In the case of 2, you have $(0,1)R(1,0)$ and $(1,0)R(0,1)$ but $(1,0)\ne (0,1)$