In the definition of partial order, notice that antisymmetric is not the oppo
site of symmetric, since a relation can be both symmetric and antisymmetric:
for example, the identity relation or the empty relation.
E
XAMPLE
5.2
1. The numerical orders
≤
on
N
,
Z
and
R
are total orders. The orders
<
are strict partial orders.
2. Division on
N \{
0
}
is a partial order:
∀
n, m
∈
N
.
n
≤
m
iff
n
divides
m
.
3. For any set
A
, the power set of
A
ordered by subset inclusion is a
partial order.
4. Suppose
(
A,
≤
A
)
is a partial order and
B
⊆
A
.
Then
(
B,
≤
B
)
is a
partial order, where
≤
B
denotes the restriction of
≤
A
to the set
B
.
5. Define a relation on formulae by:
A
≤
B
if and only if
A
→
B
. Then
≤
is a preorder. For example,
false
≤
A
≤
true
and
A
≤
A
∨
B
.
6. For any two partially ordered sets
(
A,
≤
A
)
and
(
B,
≤
B
)
, there are two
important orders on the product set
A
×
B
:
•
product order:
(
a
1
, b
1
)
≤
P
(
a
2
, b
2
)
iff
(
a
1
≤
A
a
2
)
∧
(
b
1
≤
B
b
2
)
•
lexicographic order:
(
a
1
, b
1
)
≤
L
(
a
2
, b
2
)
iff
(
a
1
<
A
a
2
)
∨
(
a
1
=
a
2
∧
b
1
≤
B
b
2
)
.
If
(
A,
≤
)
and
(
B,
≤
)
are both total orders, then the lexicographic order
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 Spring '09
 Koskesh
 Math, Order theory, Partially ordered set, Total order, partial order

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