# Question and Answers on HW#5

#### Due on paper in class, Friday 5 November 2004

Questions are in black, answers in blue

#### Question #3, 4 November 2004

In Problem 5.5.1, I think I should be using induction, The base case is that λ is in L(S)-1, right?

You are right that you should use induction. You have a predicate P(S) that says "L(S)a-1 is a regular language", and you want to prove that P(S) is true for all regular languages S. So "S = ∅" and "S = a" are the base cases, where actually you want to split the latter into "S = a" where a is the arbitrary letter in the "L(S)a-1", and "S = b" where b is any other letter. Then you will have three inductive cases for union, concatenation, and star, just as in the examples in Section 5.5. You have a typo when you say "L(S)-1", because we don't know how to take the inverse of a language. You probably meant "L(S)a-1", which is defined.

#### Question #2, 4 November 2004

Isn't there a typo on Problem 6.4.1? I think the last "j-k falling" should be "k-j falling", since the former makes no sense if j< k.

Quite right, this has been fixed.

#### Question #2, 1 November 2004

Now that Problem 5.4.2 has been corrected to (S+T)* = S*T*, I am still having trouble setting it up. Can you help?

First, because it is much easier to show a star language to be contained in another language than to show it equal directly, you should divide the problem into two parts. First show that (S+T)* is a subset of S*T*, then show the containment in the other direction. For each piece, you will want to use the inductive definition of the star language. Last modified 4 November 2004