Rob Bayer
Math 55 Review
August 13, 2009
Instructions
•
Work through the following review problems as a group
•
Make sure to focus not just on getting the correct answers, but also on how you would actually write your
proofs/solutions
•
Feel free to skip around–there’s way more problems here than the actual midterm will have, so focus on
whatever your group wants practice with.
•
As always with review/practice tests, the inclusion or exclusion of certain topics should not be taken as an
indication of what will be on the actual midterm.
•
These problems are meant to be a
supplement
to the worksheets, quizzes, and homeworks you’ve had so far.
Logic, Sets, Functions
1. Find a truth table for the compound proposition (
¬
p
→
q
)
∨
r
→
(
¬
q
∧
r
) and determine whether it is a
tautology.
2. Find the converse and the contrapositive for the statement “If it is cloudy, then it is raining”
3. Determine whether each of the following logical statements are true. The domain for all quantifiers is
N
(a)
∀
x
∃
y
(
x
≤
y
)
(b)
∃
y
∀
x
(
x
≤
y
)
(c)
∀
x
∃
y
(
y
6
=
x
∧
x
≡
y
(mod 5)
(d)
∀
x
(13
6 
x
→ ∃
y
(
xy
≡
1 (mod 13)))
(e)
∀
x
(6
6 
x
→ ∃
y
(
xy
≡
1 (mod 6)))
4. Suppose
∼
is an equivalence relation on the set
A
which has countably many equivalence classes, each of which
is countable. Show that
A
is countable.
5. Prove that if
A
is uncountable and
A
⊆
B
then
B
is uncountable.
6. Prove that log
7
15 is irrational.
7. Prove that
A

(
A

B
) =
A
∩
B
for any sets
A, B
8. Prove that if

A

=

B

, then
P
(
A
)

=
P
(
B
)

Number Theory
1. Find gcd(142
,
76) along with integers
s, t
such that 142
s
+ 76
t
= gcd(142
,
76)
2. Show that there are no solutions in positive integers to
x
2

5
y
2
= 2
3. Show that a number is divisible by 7 iff the sum of its octal (ie, base 8) digits is also divisible by 7.
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 Summer '08
 STRAIN
 Math, Equivalence relation, Transitive relation, Partially ordered set, positive integers, partial order, mod m/ gcd

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