Intro to Automata theory by Ullman Exercise 2nd

Question

Exercise 2.2.8: Let A be a DFA and a particular input symbol of A, such that for all states q of A we have delta(q,a) = q.

A)  Show that for all n >= 0, Delta cap(q, a^n) = q. where a^n is the string consisting of n a's.

note delta = δ and delta cap =  δˆ.

Comments

By seeing 'a' if u remain in same state => loop at state q

So by processing any no. of a's you remain in state q.

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ok, So n denotes the times of loop made here. if I put n = 1. delta = delta-cap. this is what your proposition? right bro.?

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Yes

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Hmm. I got it, bro. thanks, Verma Ashish. Bro, I'm solving exercises of Automata theory by Ullman as I preferred this book for my prep. 2019. Can you please let me know where can I get solution manual of this 3rd edition. that will be only for matching my solutions and some problems that I don't make through.

Hope,