M
R
∪
S
=
M
R
∨
M
R
=
(c)
Construct the matrix representing the relationT=R
1
that
is the inverse of R.
100
M
T
=
M
R
T
=
110
_________________________________________________________________
4. (10 pts.)
(a)
What is the numerical value of the postfix
expression below?
32*2
↑
5384/*
=
62
↑
=
3
65384/*
=
3
6284/*
=
3
622*
=
3
64
=3
2
(b)
What is the numerical value of the prefix expression below?
+*235/
↑
234
=
+*235/84
=
+*2352
=
+652
=
+12
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_________________________________________________________________
5. (15 pts.)
(a) Draw a directed graph G
1
whose adjacency
matrix is given on the left below.//
V
1
={a
,b
,c
,d}
.
0100
0011
0001
0000
.
(b)
Now draw the underlying undirected graph G
2
for the directed
graph G
1
of part (a) of this problem.// V
2
.
(c)
Is G
2
=(
V
2
,E
2
), above, isomorphic to the simple graph
G
3
V
3
,E
3
) given below?
Either display an isomorphism
f:V
2
→
V
3
or very briefly explain why there is no such function
by revealing an invariant that one graph has that the other
doesn’t.
Yes, define f:V
2
→
V
3
by f(a) = b, f(b) = c,
f(c) = a, and
f(d) = d.
Then
{a,
b}
ε
E
2
↔
{ f(a), f(b)}={b
,c}
ε
E
3
,
{b,
c}
ε
E
2
↔
{ f(b), f(c)}={c
,a}
ε
E
3
,
d}
ε
E
2
↔
{ f(b), f(d)}={c
ε
E
3
,
{c,
ε
E
2
↔
{ f(c), f(d)}={a
ε
E
3
.
[You could also use f(a) = b, f(b) = c,
f(c) = d, and f(d) = a.]
_________________________________________________________________
6. (10 pts.)
Recall that the composition of two relations R
and S on a set A is given by
SR={ (a,c)
ε
A×A
(
∃
b)(b
ε
A and (a,b)
ε
R and (b,c)
ε
S)}.
Also, recall that the n
th
composition power of a relation on a
set is defined recursively by R
1
= R, and for each n
ε
,i
f
n
≥
1, then R
n+1
=R
n
R.
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 Spring '08
 STAFF
 Graph Theory, Equivalence relation, Transitive relation, Tree traversal

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