(A)
Let us assume the DFA that accepts L, D(L)=(Q,Σ,q0,δ,F)
The extended transition function for DFA, D(L) is δ*
Given that ss’ ∈ L,
So as a valid string for D(L), δ*(q0, ss’) ∈ Q (To be precise F, but as F is a subset of Q, Q also works)
s ∈ L(P) so if we put s into D(L), δ*(q0, s) ∈ Q,
as Q is a finite set we can see that every string in L(P) leads us to a single state, and all the strings combined can lead us to a subset of Q
So a DFA for L(P) can be created.
(B)
L(R) can be obtained by swapping the initial and final states and reversing the edges. If there are multiple final states, first create an NFA with single final state(new final state with ∈ transitions from all final states) then convert that to a DFA.