Cartesian Product Formula Check

The question is to check if the following are true or not. When the set operations only include $\displaystyle \cup \cap$ and -, I usually draw a ven diagram, but that doesn't seem to help much for the cartesian product... what's the best way to try to tackle problems like the following

1. $\displaystyle (A\times B)\subset (C\times D)\Rightarrow A\subset C$ and $\displaystyle B\subset D$

2. $\displaystyle (A\times B)\subset (C\times D)\Rightarrow A\subset C$ and $\displaystyle B\subset D$ assuming A and B are not empty

I seem to think both are true, so I'm not sure why in the second one we put the restriction on A and B

**** I now see that if we let B be empty, then the product of A and B has to be a subset of the product of C and D, but A would not have to be a subset of C...