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COT 3100
Summer 2001
Quiz #3 (Solutions)
06/21/2001
1. If {{
a
,
c
,
e
}, {
b
,
d
,
f
}} is a partition of the set
A
= {
a
,
b
,
c
,
d
,
e
,
f
}, determine the
corresponding equivalence relation
R
.
For the relation that corresponds to the given partition, we have
A
/
R
={{
a
,
c
,
e
}, {
b
,
d
,
f
}}, so there are two equivalence classes:
[
a
]
R
= {
a
,
c
,
e
}, and [
b
]
R
= {
b
,
d
,
f
}. Using the definition of equivalence classes and
properties of equivalence relations we can find:
R
= {(
a
,
a
), (
b
,
b
), (
c
,
c
), (
d
,
d
), (
e
,
e
), (
f
,
f
),
(
a
,
c
), (
a
,
e
), (
c
,
e
), (
c
,
a
), (
e
,
a
), (
e
,
c
),
(
b
,
d
), (
b
,
f
), (
d
,
f
), (
d
,
b
), (
f
,
b
), (
f
,
d
)}
2. Consider the following relation
R
on the set
A
= {1, 2, 3, 4}:
R
= {(1, 2), (2, 3), (3, 1), (4, 1)}
Find the transitive closure of this relation.
We can find all pairs of elements in
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 Spring '09

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