Give an example of a language $L$ such that $|L| = 5$ and $|L^2| = 16$.

Question

Can a formal language $L$ have identical string in it? For example, $L =\{a, a, b, c, b, d\}$. Is $L$ valid, if yes then is $|L|= 6$ or $|L| = 4$?

Answer 1

Consider the language $L=\{a,aa,aaa,aaaa,b\}$. $$L^2=\{aa,aaa,aaaa,aaaaa,aaaaaa,aaaaaaa,aaaaaaaa,ab,ba,aab,baa,aaab,baaa,aaaab,baaaa,bb\}$$