Section 2.5
(b) Assuming it exists, the least upper bound of
A
and
B
has two properties:
(1)
A
~
L, B
~
L;
(2)
if
A
~
C
and
B
~
C,
then
L
~
C.
49
We must prove that
Au B
has these properties. Since
A
~
Au B
and
B
~
AU B, AU B
satisfies
(1). Also, if
A
~
C
and
B
~
C,
then
A
U
B
~
C,
so
A
U
B
satisfies (2) and
A
U
B
=
A VB.
13. (a) [BB]
a
V
b
=
b
and here is why. We are given
a
:5
b
and have
b
:5
b
by reflexivity. Thus
b
is an
upper bound for
a
and
b.
It is least because if c is any other upper bound, then
a
:5 c,
b
:5 c; in
particular,
b
:5 c.
(b)
a
1\
b
=
a
and here is why. We are given
a
:5
b
and have
a
:5
a
by reflexivity. Thus
a
is a lower
bound for
a
and
b.
It
is greatest because if c is any other lower bound, then c :5
a,
c :5
b;
in
particular, c :5
a.
14. (a) [BB] Suppose
x
and
y
are each glbs of two elements
a
and
b.
Then
x
:5
a, x
:5
b
implies
x
:5
y
because
y
is a
greatest
lower bound, and
y
:5
a, y
:5
b
implies
y
:5
x
because
x
is greatest. So,
by antisymmetry,
x
=
y.
(b) Suppose
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 Summer '10
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 Graph Theory, Order theory, upper bound, 5 m, Partially ordered set

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