# CMPSCI 401: Theory of Computation

### David Mix Barrington

### Spring, 2008

# Homework Assignment #4

#### Posted Monday 24 March 2008

#### Due on paper in class, Wednesday 9 April 2008

There are fifteen questions for 100 total points plus 10 extra
credit. All but the first two are from
the textbook, *Introduction to the Theory of Computation*
by Michael Sipser (second edition).
Numbers in parentheses following each problem
are its individual point value.

Students are responsible for understanding and following
the academic honesty
policies indicated on this page.

- Question X.1 (5): Ignoring the definitions in the Sipser text, define
a nonempty set A to be
*countable* if there exists a function from **N** (the
positive integers) onto A. Using this definition only, prove the following: Let
{A_{i}: i ∈ **N**} = {A_{1}, A_{2}, A_{3},...} be a collection of countable sets. Then the union of all the
A_{i} is also a countable set.
- Question X.2 (5): Let Σ = {a
_{i}: i ∈ **N**} be an
infinite alphabet. Using the definition from Question X.1, prove that
Σ^{*}, the set of all finite strings from Σ, is a countable
set.
- Problem 4.24 (10)
- Problem 4.28 (10)
- Exercise 5.1 (5)
- Exercise 5.2 (5)
- Exercise 5.3 (5)
- Exercise 5.4 (5)
- Problem 5.17 (10)
- Problem 5.19 (5)
- Problem 5.20 (10)
- Problem 5.22 (5)
- Problem 5.25 (10)
- Problem 5.35 (10 XC)
- Problem 6.7 (10)

Last modified 24 March 2008