CS 390 Test I

February 25, 1997


1. Which of the following statements are true and which are false ?
(a) If tex2html_wrap_inline62 , then tex2html_wrap_inline64 for any integer tex2html_wrap_inline66 .
(b) tex2html_wrap_inline68
(c) Let A be the domain of a function f, let tex2html_wrap_inline74 and let tex2html_wrap_inline76 .
Then tex2html_wrap_inline78 .
(d) If languages tex2html_wrap_inline80 and tex2html_wrap_inline82 are regular, then tex2html_wrap_inline84 is regular.
(e) The function tex2html_wrap_inline86 defined as tex2html_wrap_inline88 is one-to-one and onto.

2. Consider the language of strings consisting of one or more a's and zero or more b's. The alphabet for this language is tex2html_wrap_inline94 .
(a) What is the shortest string of this language ?
(b) List all the strings of length 2 of this language.
(c) Show what kind of operations are necessary to generate strings of length 3 from strings of length 2.
(d) Inductively define this language.

3. Prove by induction that tex2html_wrap_inline102 for any string x and any natural number n.

4. Find a regular expression for the language of strings over the alphabet {a,b} which are accepted by the FA given by the following transition table.

tabular29

The initial state is tex2html_wrap_inline194 and the accepting state is tex2html_wrap_inline196 .