# CMPSCI 250: Introduction to Computation

### David Mix Barrington

### Fall, 2004

# 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