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CS 173: Discrete Structures, Spring 2010
Homework 3 Solution
This homework contains 3 problems worth a total of 40 points.
When a problem speciFes a particular proof technique, you must use that technique in your
solution, even if it’s not the only reasonable approach to the mathematical problem. This is
because the main point of these problems is to learn how to use the various di±erent proof
techniques.
1.
[8 points] Thinking about negations
Using precise mathematical notation, give the negations of the following statements.
(a)
∃
!
x
∈
R
, P
(
x
)
[Solution]
This is equivalent to saying:
(
∃
x
∈
R
, P
(
x
))
∧
(
∀
y
∈
R
, P
(
y
)
→
(
y
=
x
))
The negation of which is:
¬
((
∃
x
∈
R
, P
(
x
))
∧
(
∀
y
∈
R
, P
(
y
)
→
(
y
=
x
)))
¬
(
∃
x
∈
R
, P
(
x
))
∨ ¬
(
∀
y
∈
R
, P
(
y
)
→
(
y
=
x
))
(
∀
x
∈
R
,
¬
P
(
x
))
∨
(
∃
y
∈
R
,
¬
(
¬
P
(
y
)
∨
(
y
=
x
)))
(
∀
x
∈
R
,
¬
P
(
x
))
∨
(
∃
y
∈
R
,
(
P
(
y
)
∧
(
y
n
=
x
)))
Or, in English:
P
is either not true for any real number, or true for more than
one real number.
(b)
p
∈
Z
and
m
∈
Z
and
n
∈
Z
and
p
is not the greatest common divisor of
m
and
n
[Solution]
p /
∈
Z
∨
m /
∈
Z
∨
n /
∈
Z
∨
p
=
GCD
(
m, n
)
You can use any combination of mathematical shorthand or mathematical English
(e.g., for the logical operations) but be clear about the order in which operations are
applied using parentheses, indentation or line breaks.
2.
[22 points]
A triple (
a, b, c
) of positive integers is
Pythagorean
if
a
2
+
b
2
=
c
2
.
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 Spring '08
 Erickson

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