Finite Automata Combined with Relation

Question

Let DFA , M = (Q, ∑, δ, q$_0$, F) and Relation R is defined on Q as R:Q$\rightarrow$Q such that pRq iff 

$\forall$ w ∈ $\Sigma$* [ δ*(p,w) ∈ F $\leftrightarrow$ δ*(q,w) ∈ F  OR  δ* (p, w) ∉ F $\leftrightarrow$ δ* (q, w) ∉ F] then ____________ 

A)  R is Reflexive

B) R is Symmetric

C) R is transitive

D) None

Comments

I think correction required in the question.

δ*(p,w) ∈ F need to be replaced with δ*(q,w) ∈ F before "OR".

Check and confirm.

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R(p,q) is inderctly saying,

for all w, both state p and state q are either lead to final state combinely or non-final state combinely.

That means, both p and q are equivalent states.

 

Therefore,

i) every state p is equivalent to state p. - Reflexive.

ii) if state p,q equivalent, R(p,q) and R(q,p). - Symmetric.

iii) if p,q are equivalent and q,r are equivalent then p,r also equivalent states. - Transitive.