Let M = (Q,∑, q0,A, δ) be an FA. Below are other conceivable methods of defining the extended transition function δ∗. In each case, determine whether it is in fact a valid definition of a function on the set Q × ∑∗, and why. If it is, show using mathematical induction
a. For every q ∈ Q, δ ∗ (q,) = q; for every y ∈ ∗, σ ∈ , and q ∈ Q, δ ∗ (q, yσ ) = δ ∗ (δ ∗ (q, y), σ ).
b. For every q ∈ Q, δ ∗ (q,) = q; for every y ∈ ∗, σ ∈ , and q ∈ Q, δ ∗ (q, σy) = δ ∗ (δ(q, σ ), y).
c. For every q ∈ Q, δ ∗ (q,) = q; for every q ∈ Q and every σ ∈ , δ ∗ (q, σ ) = δ(q, σ); for every q ∈ Q, and every x and y in ∗, δ ∗ (q, xy) = δ ∗ (δ ∗ (q, x), y).
I stark at the first(a) , if i need to proof in induction what is the initial step
and n step to n+1 step? can you show me the proof?
thanks!