# Homework Assignment #4

#### 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.

• 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 {Ai: i ∈ N} = {A1, A2, A3,...} be a collection of countable sets. Then the union of all the Ai is also a countable set.

• Question X.2 (5): Let Σ = {ai: 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)