CS 381 Solutions to Homework 11
Textbook pp. 563 - 564:
2
a) Equivalence relation
c) Not transitive. Hence not an equivalence relation
10. Since is an equivalence relation, equivalence classes exist
among the elements of . Let be a mapping from to the set of equivalence classes, that is . Then is a function because
for every element of , a unique exists. Also
if and only if . Hence if and only if .
12. It is reflexive because any bit string completely agrees with itself.
It is symmetric because if a string x agrees with a string y everywhere except
at the first three bits, then y agreew with x everywhere except
at the first three bits.
It is transitive because if x agrees with y and y agrees wtih z
everywhere except
at the first three bits, respectively, then x agrees with z everywhere except
at the first three bits.
40 a) { (x, 3x/2) | x is a positive even integer}
44
b) Not a partition because 0 is not included.
d) Yes, it is a partition.
Textbook pp. 578 - 580:
10. Not a partial order because it it not transitive. is missing.
20. The vertices are 0, 1, 2, 3, 4, 5 from the top and the arcs are (5,4), (4,3),
(3,2), (2,1) and (1,0).
34. a) 27, 48, 60 and 72
b) 2 and 9
c) No greatest element
d) No least element
e) 18, 36 and 72
f) 18
g) 2, 4, 6 and 12
h) 12