Consider the language A$2\{0,1\}$

$$ defined as

A$2=\{$ w $|$ w begins and ends with the same symbol. $\}$

Therefore $010$ in A$2$ whereas $110$ in $/$ A$2.$ Also, since the string does not have a beginning or ending

symbol, $/$ in A$2.$

Suppose that we define A$3=$ A$2\backslash$cup $\{\},$ i$.$e$.,$ the set of all the strings that begin and end with the

same symbol and the string $.$ What is the smallest lower bound on the number of states for the

DFA that recognizes the language A$3.$