This preview shows page 1. Sign up to view the full content.
Section 2.5
47
(c) This is not a partial order because the relation is not antisyrrunetric; for example,
3
j
3 because
(_3)2
~
3
2
and 3
j
3 because 3
2
~
(_3)2 but
3 ::j:. 3.
(d) This is not a partial order because the relation is not antisyrrunetric; for example, (1,4)
j
(1,8)
because 1
~
1 and similarly, (1,8)
j
(1,4), but (1,4) ::j:. (1,8).
(e) This is a partial order.
Reflexive:
For any
(a, b)
E
N x N,
(a, b) j (a, b)
because
a
~
a
and
b
~
b.
Antisymmetric:
If
(a, b), (c, d)
E
N x N,
(a, b) j
(c, d)
and
(c, d) j
(a, b),
then
a
~
c,
b
~
d,
c
~
a
and
d
~
b.
So
a
=
c,
b
=
d
and, hence,
(a, b)
=
(c, d).
Transitive:
If
(a, b), (c, d), (e,
f)
E
N x N,
(a, b) j (c, d)
and
(c, d) j (e,
f), then
a
~
c,
b
~
d,
c
~
e and
d
~
f.
So
a
~
e (because
a
~
c
~
e) and
b
~
f
(because
b
~
d
~
f)
and, therefore,
(a, b) j (e,
f).
This is not a total order; for example, (1,4) and (2,5) are incomparable.
(t) This is reflexive and transitive but not antisymmetric and, hence, not a partial order. For example,
This is the end of the preview. Sign up
to
access the rest of the document.
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

Click to edit the document details