Homework #4
Solutions
Due: October 4, 2010
1. For each of the following relations
R
on the set
X
determine whether
R
is re±exive,
symmetric, antisymmetric, or transitive.
(a)
X
=
Z
aRa
if and only if
a
±
b
+ 1;
(b)
X
=
Z
aRb
if and only if
a
+
b
is even;
(c)
X
=
Z
aRb
if and only if
a
+
b
is odd;
(d)
X
=
Z
aRb
if and only if
a
and
b
are relatively prime;
(e)
X
=
R
aRb
if and only if
a
²
b
is an integer;
(f)
X
=
R
aRb
if and only if
a
2
±
b
2
.
You may record your answers (yes/no) in the following table:
Relation
Re±exive
Symmetric
Antisymmetric
Transitive
(a)
yes
no
no
no
(b)
yes
yes
no
yes
(c)
no
yes
no
no
(d)
no
yes
no
no
(e)
yes
yes
no
yes
(f)
yes
yes
no
yes
I
Solution.
Some comments on the answers recorded in the table:
(a)
Re±exive
:
a
±
a
+ 1 is always true, so
R
is re±exive.
Symmetric
: Let
a
= 0,
b
= 2. Then 0
±
2 + 1 but 2
6±
0 + 1. That is,
aRb
but
b
is not related to
a
, so
R
is not symmetric.
Antisymmetric
: Let
a
= 1,
b
= 0. Then
a
±
b
+ 1 so
aRb
and
b
±
a
+ 1 so
bRa
, but
a
6
=
b
, so the relation is not antisymmetric.
Transitive
: Let
a
= 1,
b
= 0, and
c
=
²
1. Then
a
±
b
+ 1 and
b
±
c
+ 1, but
a
±
c
+ 1, so the
relation is not transitive.
(b)
Re±exive
: 2
a
is always even, so
R
is re±exive.
Symmetric
:
a
+
b
=
b
+
a
so
R
is symmetric.
Antisymmetric
;
a
+
b
=
b
+
a
even does not imply that
a
=
b
, so
R
is not antisymmetric.
Transitive
: If
a
+
b
and
b
+
c
are both even, then so is
(
a
+
b
) + (
b
+
c
)
²
2
b
=
a
+
c
, so
R
is transitive.
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 Fall '09

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