MATH 433 March 2, 2015 Quiz 5: Solutions

MATH 433
March 2, 2015
Quiz 5: Solutions
Problem 1. Let R be the relation defined on the set of positive integers by xRy if and only
if x = y n for some positive integer n (“is a power of”). Is this relation reflexive? Symmetric?
Transitive? Explain how you know.
Solution: the relation R is reflexive and transitive, but not symmetric (actually, it is weakly
Any positive integer x is related to itself by R since x = x1 . Hence R is reflexive.
The relation R is not symmetric as the following counterexample shows: 4R2 as 4 = 22 , but 2 is not
related to 4 as 2 = 41/2 .
Assume that xRy and yRz for positive integers x, y, an z. Then x = y n1 and y = z n2 , where n1 , n2 ∈ Z,
n1 , n2 > 0. It follows that x = (z n2 )n1 = z n2 n1 so that xRz. Thus R is transitive.
Problem 2. Construct an acceptor automaton over the alphabet {a, b} which accepts those
input words that do not contain a subword ab (and rejects any input word containing a subword ab).
Solution: The smallest automaton performing the task has 3 states:
a, b