Math 135  Midterm Examination
1
Math 135
Algebra
Midterm Solutions
Question 1. (a)
[4 marks] Consider the following two statements:
∃
x
∃
y
(
P
(
x
) =
⇒
Q
(
y
))
,
and
(
∀
x P
(
x
)) =
⇒
(
∃
y Q
(
y
))
Give an example of a universe of discourse, and meaning for
P
and
Q
for which the
frst
statement
is
true
. Give an example where the
second
statement is
False
.
Solution:
There are many possible correct answers to this question. Here’s one:
For the positive integers, and
P
(
x
) =
Q
(
x
) = “
x <
1”, the ±rst statement is true.
For the positive integers,
P
(
x
) = “
x
≥
1”, and
Q
(
y
) = “
y <
1”, the second statement is false.
(b)
[4 marks] Negate the two statements given in part (a) above, and simplify them so that no
quanti±er or compound statement is negated. Show the steps in the simpli±cation.
Solution:
¬
(
∃
x
∃
y
(
P
(
x
) =
⇒
Q
(
y
))) is equivalent to
∀
x
∀
y
¬
(
P
(
x
) =
⇒
Q
(
y
))
∀
x
∀
y
(
P
(
x
))
∧ ¬
Q
(
y
))
.
¬
((
∀
x P
(
x
)) =
⇒
(
∃
y Q
(
y
))) is equivalent to
(
∀
x P
(
x
))
∧ ¬
(
∃
y Q
(
y
))
(
∀
x P
(
x
))
∧
(
∀
y
¬
Q
(
y
))
.
(c)
[4 marks] Are the two statements given in part (a) above equivalent?
Justify your answer
appropriately.
Solution:
The two statements are equivalent.
There are several ways of proving this.
The easiest is to prove that the negations of the two
statements we obtained in part (b) above are equivalent. We prove this below.
Let
S
1
be the negation of the ±rst statement,
S
2
that of the second. Suppose
S
1
is true. Then, for
every pair of elements
x, y
in the universe,
P
(
x
) is true and
Q
(
y
) is false. This means that
∀
x P
(
x
)
is true, and
∀
y
(
¬
Q
(
y
)) is also true. This implies that
S
2
is true.
Conversely, suppose that
S
2
is true. Then, the two statements
∀
x P
(
x
) and
∀
y
(
¬
Q
(
y
)) are both
true. Thus, for every pair
x, y
in the universe,
P
(
x
)
∧ ¬
Q
(
y
) is true. In other words,
S
1
is true.
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View Full DocumentMath 135  Midterm Examination
2
Question 2. (a)
[4 marks] Factorize the two integers 11571 and 9338 into a product of primes, and
compute their greatest common divisor. Explain the method you used to factorize the numbers,
and show your steps.
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 Spring '11
 Phystago
 Math, Algebra, Natural number, Prime number, Euclidean algorithm, gcd

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