# CMPSCI 501: Theory of Computation

### David Mix Barrington

### Spring, 2014

# Homework Assignment #1

#### Posted Monday 27 January 2015

#### 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.

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

- Problem A-1 (10): Let A be any nonempty set. A
**partition** of A is
a collection P = {S_{1},..., S_{k}} 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 2
^{n.
} - (b, 10) The number of binary strings of length 6n, with a
number
of 1's divisible by 3, is exactly (2
^{6n} + 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
{a
_{1},...,a_{n}}
such that 1/a_{1}^{2} + ... +
1/a_{n}^{2} = 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.

Last modified 2 February 2015