Is this the correct answer for the cardinality of this set?
This is how you should solve it.
Let $\Sigma_1=\Sigma$
We are asked to find: $|\Sigma^ 4|$
Now
$$\Sigma^4 = \Sigma . \Sigma . \Sigma . \Sigma = \{a,b\} . \{a,b\}. \{a,b\} . \{a,b\}$$
$$\text{ Here . means the concatenation operator}$$
So this is equivalent to number of ways of forming strings of length $4$ with symbols from $\Sigma$.
We have $4$ places,

Each of the places have two choices to get filled : either $a$ or $b$.
So,
_   _  _   _
2 . 2. 2. 2

So answer is $2^4 =16$