Minimum No of states in DFA

Question

No. of states in the DFA accepting the following set of strings are:

( ( aa* + φ* )* (aa* + φ* ) + bb* + φ* φ + φ* )*

Quite confusing to me. Share your approach!

Comments

( ( aa* + φ* )* (aa* + φ* ) + bb* + φ* φ + φ* )*

 ( aa* + φ* )*

 we know φ*=∈
 so now it become ( aa* + ∈ )* =a* 

similarly we have  ( bb* + φ* )*=b*
  

so now it become 

( ( a* )* (a*) + b* + φ* φ )*
now we know ∅* =∈
( ( a* )* (a*) + b* + ∈ φ)*

( ( a* )* (a*) + b* + ∈ φ )*

( ( a* )* (a*) + b* + ∈ φ )*
so here ∈ φ= ∅
( ( a* )* (a*) + b* +∅ )*
 

(  a* + b* +∅ )*   {minimal form of above question}

1 state is sufficient i think :)
 

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aa* + $\Phi$*

= a⁺ + $\epsilon$

= a*

bb* + $\Phi$*$\Phi$ + $\Phi$*

= b⁺ + $\epsilon$$\Phi$ + $\epsilon$

= b⁺ + $\Phi$ + $\epsilon$

= b⁺ + $\epsilon$

= b*

Thus, answer = (a*a* + b*)*

 = (a* + b*)*

= (a+b)*

So, 1 state should be sufficient

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@Kunal. If it accepts $\phi$, then 1 state isnt sufficient.

Answer 1

( ( aa* + φ* )* (aa* + φ* ) + bb* + φ* φ + φ* )*

=((a⁺ + epsilon)* (a⁺+epsilon) + b⁺ + epsilon)*
=(a*a* + b*)*
=(a*+b*)*
=(a+b)*
only 1 state needed