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Important question for exam

Consider the following machine. Select correct option.

Where  ${\mathbf{L}}_{\mathbf{B}}$ = The all strings accepted at B state, if B is final. 

${\mathbf{L}}_{\mathbf{C}}$ = The all strings accepted at C state, if C is final.

Answer: 

We can conclude the relation between ${\mathbf{L}}_{\mathbf{B}}$ and ${\mathbf{L}}_{\mathbf{C}}$ fast by Arden’s Theorem i.e. if B is final then Language of this machine is ${\mathbf{L}}_{\mathbf{B}}$ and if C is final then Language of this machine is ${\mathbf{L}}_{\mathbf{C}}$.

We consider that B is final state then given language is ${\mathbf{L}}_{\mathbf{B}}$. 

Now we reach state B by D.0 or B.1 or C.0. 

Same way we can calculate ${\mathbf{L}}_{\mathbf{C}}$ for state C. 

Now if we see properly ${\mathbf{L}}_{\mathbf{B}}$ and ${\mathbf{L}}_{\mathbf{C}}$ is same.

So, ${\mathbf{L}}_{\mathbf{B}}$ = ${\mathbf{L}}_{\mathbf{C}}$, hence option c is correct.