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
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 Fall '09
 Equivalence relation, Binary relation, Transitive relation

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