Math 552 Final Exam SOLUTIONS
Rob Bayer
August 14, 2009
You have until 4:00pm to complete this test. No calculators, books, notes, or consultation with other members of
the class are permitted. Your exam should have 4 pages.
Unsupported or improperly supported answers will receive no credit.
1. (10 pts) Short Answer. You need not show any work for this problem
(a) A function
f
is injective iff
f
(
x
) =
f
(
y
)
⇒
x
=
y
(b) A relation
R
is an equivalence relation iff it is symmetric, transitive, reflexive
(c) Two events are called independent iff P(E —F) = P(E)
(d) Circle the countable sets:
R
Q
Z
×
N
P
(
N
)
(e) The linear congruence
ax
≡
b
(mod
n
) has a solution iff gcd(
a, n
)

b
(f) A function
f
:
G
1
→
G
2
is an isomorphism of graphs iff
f
is bijective and there is an edge b/w
u
and
v
in
G
1
iff there is an edge b/w
f
(
u
) and
f
(
v
) in
G
2
(g) State Chebyshev’s Inequality
P
(

X

E
[
X
]
 ≥
r
)
≤
V
(
X
)
/r
2
(h) Find a Minimal Spanning Tree in the following graph. You may simply fill in the necessary edges
graphsol.pdf
(i) Find a generating function for the number of solutions in nonnegative integers to 2
x
1
+ 3
x
2
+ 4
x
3
=
n
1
1

x
2
1
1

x
3
1
1

x
4
2. (15 pts)
(a) Find a truth table for the compound proposition (
p
∨
q
)
→
(
q
∧ ¬
r
)
p
q
r
p
∨
q
q
∧ ¬
r
(
p
∨
q
)
→
(
q
∧ ¬
r
)
T
T
T
T
F
F
T
T
F
T
T
T
T
F
T
T
F
F
T
F
F
T
F
F
F
T
T
T
F
F
F
T
F
T
T
T
F
F
T
F
F
T
F
F
F
F
F
T
1
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(b) Prove or provide a counterexample:
A

(
B

C
) = (
A

B
)

C
for any sets
A, B,
and
C
This is false. If
A
=
B
=
C
=
{
1
}
, then
A

(
B

C
) =
A
 ∅
=
A
=
{
1
}
and (
A

B
)

C
=
∅ 
C
=
∅
3. (15 pts)
(a) Find a solution to 63
x
≡
3 (mod 132)
We start wit hthhe extended Euclidean Algorithm:
132 = 2
·
63 + 6
63 = 10
·
6 + 3
6 = 2
·
3 + 0
Working backwards, we get
3 = 63

10
·
6
= 63

10
·
(132
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 Summer '08
 STRAIN
 Math, Graph connectivity, connected component, Strongly connected component, ith flip

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