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EECS 203: Homework 10 Solutions
Section 8.4
1. (E) 2
When we add the pairs
(
x,x
)
to the given relation we have all of
Z
×
Z
; in other words, we have the relation
that always holds.
2. (E) 6
We form the reﬂexive closure by taking the given directed graph and appending loops at all vertices at
which there are not already loops.
a
c
b
d
3. (E) 14
Suppose that the closure
C
exists. We must show that
C
is the intersection
I
of all the relations
S
that
have property
P
and contain
R
. Certainly,
I
⊂
C
, since
C
is one of the sets in the intersection. Conversely,
by deﬁnition of closure,
C
is a subset of every relation
S
that has the property
P
and contains
R
; therefore
C
is contained in their intersection.
4. (M) 22
Since
R
⊂
R
*
, clearly if
4 ⊂
R
, then
4 ⊂
R
*
.
5. (C) 24
It is certainly possible for
R
2
to contain some pairs
(
a,a
)
. For example , let
R
=
{
(1
,
2)
,
(2
,
1)
}
6. (M) 26
(a) We show the various matrices that are invoked. First,
A
=
0
0
1
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 Spring '07
 YaoyunShi

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