Consider the following sets: $A=\{0,1\}$, $B=\{\{0,1\}\}$ and $C=A\cup B$. Enumerate the following sets and report their cardinality:
No. 1: $2^C$
No. 2: $C\times C$ (cross-product)
No. 1: $\{\},\{0\},\{0,1\},\{1\},\{0,0\},\{0,1\},\{1,0\},\{1,1\}$, so we have $8$, so $2^8$ would be $256$ and that would be the cardinality.
No.2: $\{0\},\{0,1\},\{1\},\{0,0\},\{0,1\},\{1,0\},\{1,1\},\{0\},\{0,1\},\{1\},\{0,0\},\{0,1\},\{1,0\},\{1,1\}$, so it would be $14$ for the cardinality.
I just want to make sure if these are right, and if not, find out where I went wrong.
There are a number of issues here. For example, you have counted a few elements twice in your enumeration of $2^C$ e.g. $\{0,0\}=\{0\}$ and $\{0,1\}=\{1,0\}$ and missed out $B=\{\{0,1\}\}$ as a subset of $C$.
To find the power set, it helps to first explicitly write out the elements of the original set: $$C=A\cup B=\{0,1\}\cup\{\{0,1\}\}=\{0,1,\{0,1\}\}.$$ Then we count the subsets in increasing order of cardinality as $$\{\},\{0\},\{1\},\{\{0,1\}\},\{0,1\},\{0,\{0,1\}\},\{1,\{0,1\}\},\{0,1,\{0,1\}\}.$$
You can determine the cardinality of $2^C$ by noting that since the cardinality of $C$ is $3$, then the cardinality of $2^C$ is $2^3=8$.
To enumerate the elements of $C\times C$, we list the ordered pairs of elements in $C$ i.e. $$\{0,0\},\{0,1\},\{0,\{0,1\}\},\{1,0\},\{1,1\},\{1,\{0,1\}\},\{\{0,1\},0\},\{\{0,1\},1\},\{\{0,1\},\{0,1\}\}.$$ The cardinality of $C\times C$ is then $3\times 3=9$.
I assume that the purpose of this exercise is to distinguish between elements of a set and its subsets. You have to be very careful sometimes. :)