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Section 2.3
39
10. (a)
y
x
(b) The relation is not reflexive because.
for example. (2,2)
fj
n.
It
is not tran
sitive because. for example. (2,0)
En
and (0, 1) E
n
but (2, 1)
fj
n.
(c) The relation is symmetric since if
(x, y)
E
n.
then 1
~
Ixl
+ Iyl
~
2.
so 1
~
Iyl
+ Ixl
~
2. so
(y,x)
E
n.
It
is not arttisymmetric since. for exam
ple; (0,1) E
nand (1,0) E
n.
but
°
¥=
1.
11. (a) [BB] Reflexive: For any set
X.
we
ha~
X
~
X.
Not symmetric: Let
a, b
E
S.
Then
{a}
~
{a, b}
but
{a, b}
<Z
{a}.
Antisymmetric:
If
X
~
Yand Y
~
X.
then
X
=
Y.
Transitive:
If
X
~
Y
and
Y
~
Z.
then
X
~
Z.
(b) Not reflexive: For no set
X
is it true that
X
~
X.
Not symmetric: As before.
.
Antisymmetric "vacuously":
It
is impossible for
X
~
Y
and
Y
~
X.
(Recall that an implication
is false only when the hypothesis is true and the conclusion is false.)
Transitive: As before.
.
(c) Not reflexive: Since
S
¥=
0, there is some element
a
E
S,
and so some set
X
=
{a}
¥=
0
E
P(S).
For this
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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