Homework Assignment #1

Due on paper in class, Friday 6 February 2015

There are thirteen questions for 100 total points plus 15 extra credit. All but three are from the textbook, Introduction to the Theory of Computation by Michael Sipser (second edition). though some are adapted as indicated. The number in parentheses following each problem is its individual point value.

• Problem A-1 (10): Let A be any nonempty set. A partition of A is a collection P = {S1,..., Sk} of subsets of A, such that every element of A is in exactly one set of P. An equivalence relation on A is defined in the text. If R is an binary relation on A, and a is an element of A, the equivalence class of a, written [a], is the set {b: R(a, b)}. Prove that the set of equivalence classes {[a]: a ∈ A} is a partition of A if and only if R is an equivalence relation.

• Problem A-2 (15): Prove the two following statements for all nonnegative integers n by ordinary induction on n.
• (a, 5) The number of binary strings of length n+1, with an even number of 1's, is exactly 2n.
• (b, 10) The number of binary strings of length 6n, with a number of 1's divisible by 3, is exactly (26n + 2)/3, and of the remaining strings, exactly half have a number of 1's congruent to 1 modulo 3.

• Problem A-3 (15XC): (Thanks to Bill Gasarch.) Prove that for any integer n with n ≥ 6, there exist a set of exactly n positive integers {a1,...,an} such that 1/a12 + ... + 1/an2 = 1. (Hint: Use a variant of mathematical induction.)

• Exercise 1.4 parts e and f only (10).

• Exercise 1.6 parts e and h only (5).

• Exercise 1.7 parts g and h only (5).

• Exercise 1.12 (10).

• Exercise 1.18 parts e and h only (5). You may use any valid method.

• Exercise 1.28 part b only, but also produce a DFA for the language. You may use any valid method (5)

• Problem 1.31 (10). There are at least two ways to prove this -- by induction on all regular expressions, or by arguing about NFA's. For the latter, you may use the result of Exercise 1.11 (solved in the back) without proof.

• Problem 1.37 (10).

• Problem 1.38 (10).

• Problem 1.49 part a only (5). Note that this language is the set of such strings that can be written this way, for any possible k with k ≥ 1.