Prof. Bjorn Poonen
October 16, 2001
MATH 55 MIDTERM SOLUTIONS (yellow)
(1)
(5 pts. each) For each of (a)-(g) below: If the proposition is true, write TRUE.
If the proposition is false, write FALSE. (Please do not use the abbreviations T and
F, since in handwriting they are sometimes indistiguishable.) No explanations are
required in this problem.
(a) The set
{
0
,
1
}
*
of bit strings of ﬁnite length is a countable set.
TRUE, because one can list all the bit strings in a sequence, by listing all bit
strings of length zero, then all of length one, and so on:
λ,
0
,
1
,
00
,
01
,
10
,
11
,...
Alternatively,
{
0
,
1
}
*
can be viewed as a subset of the set of all ﬁnite strings of
typewriter symbols, which was shown in class to be countable.
(b) The proposition
∃
x
∀
y
(
x
≥
y
)
is true, when the universe of discourse is the
set of natural numbers.
FALSE. In words, this says “There is a natural number
x
that is greater than or
equal to all natural numbers
y
,” which is false.
Alternatively, we can prove the negation,
∀
x
∃
y
(
x < y
), as follows. Suppose that
x
is a natural number. Then there exists a natural number
y
such that
x < y
,
namely
y
=
x
+ 1.
(c) The numbers
34
,
35
,
36
are pairwise relatively prime.
FALSE, because gcd(34
,
36) = 2.
(d) There are inﬁnitely many integers
x
satisfying both
x
≡
12 (mod 99)
and
x
≡
16 (mod 100)
. (Hint: you don’t need to solve this system.)
TRUE. Since gcd(99