Ive been struggling to answer this question. The question is we have A = {[a],[b],[c]} and I want to know how many strings are in A^4 ( A to the power of 4). And also if one of the strings in A was empty how many strings would there be in A^4, for example: A= {[a],[b],[]} ?
Now I understand that A^n means A n times. So in this case A^4 means AAAA. But how do we know how many strings there are.
When $A=\{[a],[b],[c]\}$ then $A^4$ is the set of strings that have length $4$ and all characters are $a$, $b$ or $c$. This means that for every character there are $3$ options. Since there are $4$ characters, this gives $3^4=81$ options in total.
If we have $A=\{[a],[b],[]\}$ then $A^4$ is the set of all strings that have length at most $4$ and all characters are $a$ or $b$. This means that for every character there are $2$ options. We can consider all possible lengths of strings in $A^4$, which are $0$, $1$, $2$, $3$ or $4$. We find that $A^4$ has $1+2+2^2+2^3+2^4=31$ strings in total.