, is just the inductive hypothesis that you're assuming to help you solve the induction right?
you assume that a string x with a length of n holds the property , so therefore in your inductive proof that you show on your first post is the induction for n+1 and so on?
one last question, what would be the base case here? Is the base case for part a and b the same? From my point of view, we break w = ua. then assume that n = 0, and therefore u is a string of length 1 and so is a. So (au)^R is equal to au^R.
Please do correct me, as that's the way I see it
It seems the base case is intended to be a string of length 0. (cf. Cond.1)
a. Prove (xy)^R=y^Rx^R where the length of y is 0.
Use y=y^R=empty set.
b. Prove (w^R)^R=w where the length of w is 0.
Use w=w^R=empty set.
The base case is not exactly "the same" since you prove something different. But you use the same fact (that is, Cond.1)...