dfa design

Question

Design a DFA over w ∈ {a,b}^* such that number of a = 2 and there is no restriction over length of b.
Answer
We will explain this intuitive approach
Step 1 Make smallest string DFA:
Means for number of a’s = 2 , crate a DFA then in next step we will take care of b’s.

Step 2 
Now we will take care of b’s 

Can anyone explain as in this  w =  {a,b}* is   given so we have to accept null string too ???? in dfa and we should make A as also final

Comments

Tell me one think
DO you want to accept string having zero 'a's ( bbbbbbbbbbb )?

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no. of should be exactly two it should not contain 0 a , but i read that {a,b}* will contain all strings including null too , in question i think they should mention {a,b} rather than {a,b}*

Answer 1

DFA designed by you is correct,
w ∈ {a,b}^* means string can contain anything means string can be {Ɛ, a, b ,ab, ba , aa, bb,...}  =(a+b)^*

DFA accepts only those string that contains exactly two a.

Comments

oh i got my mistake thanks understood ,  i thought due to {a,b}* you have to include null in dfa , it depends on condition  ok ok :)