CS 173: Discrete Structures, Spring 2010
Homework 11
This homework contains 3 problems worth a total of 42 regular points and 4 bonus points. It
is due on Friday, April 30th at 4pm. Put your homework in the appropriate dropbox in the
Siebel basement.
1.
Partial Orders [16 points]
(a) Suppose that
A
is the set of ﬁnite nonzero length
sequences of integers
. E.g. 5
,
2
is an element of
A
. So is 3
,
4
,
3
,
3, and so is 3
,
1
,

37
,
67
,
0
,
3. A generic element of
A
would be
x
1
,x
2
,...,x
n
where
n
≥
1 (since each sequence contains at least one
integer.) Let
S
be the “subsequence of” relation on
A
deﬁned as follows:
x
1
,...,x
m
S y
1
,...,y
n
if and only if
m
≤
n
and there is an integer
k
such that
x
1
=
y
k
, x
2
=
y
k
+1
, ..., x
m
=
y
k
+
m

1
.
Prove that
S
is a partial order.
(b) Suppose you are given two sets,
X
and
Y
with partial orders
±
X
and
±
Y
. We deﬁne
a new relation
±
X
×
Y
on
X
×
Y
(the Cartesian product of sets X and Y) such that:
(
a,b
)
±
X
×
Y
(
c,d
) if and only if the following is true: