Show that there is a language A $\backslash$subseteq $\backslash \{0,1\backslash \}^*$A$\{0,1\}$ with the following properties: For all x$\backslash$in Ax in A$,|$x$|\backslash$leq $$x$<=5.$ And that no DFA with fewer than $9$ states decides A$.$ Hint: You don't have to define A explicitly; just show that it has to exist. Count the number of languages satisfying $(1)$ and the number of DFAs satisfying $(2),$ and argue that there aren't enough DFAs to decide all those languages. To count the number of languages satisfying $(1),$ think about writing down all the strings of length at most $5,$ and then to define such a language, you have to make a binary decision for each string about whether to include it in the language or not. How many ways are there to make these choices? To count the number of DFAs satisfying $(2),$ consider that a DFA behaves identically even if you rename all the states, so you can assume without loss of generality that any DFA with k states has the state set $\{$q $1,$q $2,...,$q k $\}.$ Now think about how to count how many ways there are to choose the other parts of the DFA. Number of languages satisfying $(1)?$ Number of DFAs satisfying $(2)$:$?$