MATHEMATICS 109
NAME: _______________________
February 1, 2006
Section: _______________________
FIRST
MIDTERM
EXAMINATION
There should be no books and no notes.
There should be no calculators and no cell phones.
Give complete proofs written in complete English sentences, as far as practical. If a question
is unclear or is incorrect, please ask the TA or Instructor about it.
Do all four problems.
1. (30%) For this problem, the universe of discussion is the set of real numbers.
Consider
the following statement:
(
29
(
29
(
29
xy
y
x
y
x
2
2
2
+
2200
5
.
(a) Using logical symbols write the denial of the statement a few times, each time moving the
negation sign farther to the right until the final statement has no negation in it.
(
29
(
29
(
29
[
]
xy
y
x
y
x
2
2
2
+
2200
5
¬
(
29
(
29
(
29
[
]
xy
y
x
y
x
2
2
2
+
2200
2200
¬
(
29
(
29
(
29
[
]
xy
y
x
y
x
2
2
2
+
5
2200
¬
(
29
(
29
(
29
xy
y
x
y
x
2
2
2
≤
+
5
2200
(b) Write the original statement and the last negation in English.
Original: There exists a real number
x
such that for every real number
y
,
x
2
+
y
2
> 2
xy
.
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 Winter '06
 Knutson
 Math, Rational number, Irrational number

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