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Prof. Allen Knutson’s Math 109 Midterm #2, Nov 13, 2006
Name:
SID:
If you have questions, come and ask me! I really really mean it!!!
1.
For each of the relations listed below, write
R
,
S
,
T
if the relation is
r
e±exive (
∀
x, x
∼
x
),
s
ymmetric (
∀
x, y, x
∼
y
=
⇒
y
∼
x
), or
t
ransitive
(
∀
x, y, z, x
∼
y
y
∼
z
=
⇒
x
∼
z
).
You aren’t required to prove that the
relations have these properties.
For each such property that a relation does
not
have, give an example showing
that the relation lacks the property. (That is to say, you
are
required to prove
that they
don’t
have those properties.) Part A is done as an example:
A. Let
X
=
R
, and write
a
∼
b
if
a
≤
b
.
R, T.
This
∼
is not symmetric because
55
∼
89
but
89
6
∼
55
.
B. Let
X
=
R
, and write
a
∼
b
if
b

a
is an integer multiple of
2π
.
R,S,T.
(Some people seemed to think that
0
is not an integer multiple
of
2π
, but it is.)
C. Let
X
=
R
, and write
a
∼
b
if
a
is a realnumber multiple of
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This note was uploaded on 04/07/2008 for the course MATH 109 taught by Professor Knutson during the Winter '06 term at UCSD.
 Winter '06
 Knutson
 Math

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