No, because both languages are different so you can't draw single DFA for both languages.
An automaton uniquely defined a language, but yes of-course for a language more than one automata are possible called 'equivalent automata'.
Language L1 = A = {xy | Na(x) = Nb(y)}
is a regular language. Regular expression for this language is:
(a + b)*a(a + b)*b(a + b)* + ^
To understand this language and regular expression read: "Show that the following set over {a, b}
is regular".
Language L2 = A = {w | Na(w) and Nb(w) are even number}
is also a regular language. Regular expression for this language is:
((a + b(aa)*ab)(bb)*(ba(aa)*ab(bb)*)*a + (b + a(bb)*ba)(aa)*(ab(bb)*ba(aa)*)*b)*
To understand this language and regular expression read: "Need Regular Expression for Finite Automata".
But both languages are not equal because there are some strings in language L1 those are not belongs to language L2 e.g. ab
is a string in L1 but doesn't not consist of even number of a
and b
hence doesn't belongs to language L2.
Note: Language L2 is either not a subset of language L1, because in L2 a strings of even length and single symbol is possible like aa
, aaaa
, bb
, bbbb
but these strings are not member in L1.
Both languages are different hence single DFA is not possible for both languages.