Math 240, Quiz 10
Name:
Circle One: T 12:05
T 2:25
R 12:05
R 2:25
Instructions: Answer all questions fully, showing work where necessary.
1) Let
R
be the relation on the set of ordered pairs of positive integers such
that ((
a, b
)
,
(
c, d
))
∈
R
if and only if
ad
=
bc
. Show that
R
is an equivalence
relation.
Cleary,
ab
=
ab
so that (
a, b
)
R
(
a, b
).
Also, if
ad
=
bc
then
cb
=
da
, so that
if (
a, b
)
R
(
c, d
) then (
c, d
)
R
(
a, b
).
Transitivity is the hardest:
if (
a, b
)
R
(
c, d
)
and (
c, d
)
R
(
x, y
), we need to show that (
a, b
)
R
(
x, y
), that is, we need to show
ay
=
xb
. We know that
ad
=
bc
and
cy
=
dx
. Multiplying the left sides and the
right sides, we get that
adcy
=
bcdx
. Since
dc
is present in both sides, we can
remove it to get that
ay
=
bx
.
2) Answer these questions for the poset (
{
2
,
4
,
6
,
9
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 Fall '08
 Miller
 Math, Integers, Order theory, upper bound, Partially ordered set, Upper Bounds, van vleck

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