This preview shows pages 1–2. Sign up to view the full content.
Handout #22
CS103
April 19, 2010
Robert Plummer
Problem Set #4—Due Monday, April 26 in class
1.
Determine whether the relation R on the set of all integers is reflexive, symmetric,
antisymmetric and/or transitive, where (x,y) is in R if and only if:
a) x = y+1 or x = y1
b) x is a multiple of y
c) x and y are both negative or both nonnegative
d) x = y
2
e) x
≥
y
2
2.
Suppose that R and S are reflexive relations on a set A.
Prove or disprove each of the
following statements:
(a)
R
∪
S is reflexive
(b)
R
∩
S is reflexive
(c)
R – S is irreflexive
3.
It was noted in Handout #17 that an equivalence relation partitions a set into disjoint,
nonempty subsets (p. 5).
Those subsets are called equivalence classes, as defined in
class on 4/19.
Suppose R is an equivalence relation on set A, and that a and b are elements of A.
Prove
that the following three statements are equivalent:
(i)
aRb
(ii)
[a]
= [b]
(iii)
[a]
∩
[b]
≠
∅
4.
Let R
1
and R
2
be equivalence relations on A.
Show that R1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '09

Click to edit the document details