I'm just doing some practice problems to identify sample spaces. I Need help knowing if my logic is right
a) What is the sample space for a single roll if their dice are painted two different colors?
My best guess was that the Sample space is $S=\{\{x_j|1\le j\le6\} \cup\{x_k|1\le k\le6\}\}$ where the $\{x_j|1\le j\le6\}$ is the case where the first friend rolls with his unique die first and the $\{x_k|1\le k\le6\}$ is the case where the second friend rolls with his unique die first.
b) What's the sample space for a single roll if their dice are the same color? Does this change anything?
I believe that since they're the same color it's just
$S=\{x_j|1\le k \le 6\}$ since they're the same color they're no second color/case to consider, so that's the difference.
c) What if one friend rolls both dice (I assume that the dice are distinct, it doesn't specify). Does this change anything?
I think the answer is $S =\{(x_j,x_k)|1\le x_j,x_k \le6\}$ Since it's just one friend rolling both dice for them and since they're distinct dice, order matters thus we should consider all the different possible $(x_j,x_k)$ outcomes.
Do I have this right? Please let me know if I've misinterpreted anything. Thank you for any help.