Prof. Bjorn Poonen
October 5, 2001
MATH 55 PRACTICE MIDTERM
Do not write your answers on this sheet.
Instead please write your name, your
student ID, your TA’s name, your section time, and all your answers in your blue
books.
In general, you must show your work to get credit.
Total:
100 pts., 75
minutes.
(1)
For each of (a)(g) below: If the proposition is true, write TRUE. If the propo
sition 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 circle
{
(
x, y
) :
x
∈
R
, y
∈
R
,
and
x
2
+
y
2
= 1
}
is a countable set.
(b) The propositions
p
→ ¬
q
and
q
→ ¬
p
are logically equivalent.
(c) If
p
is a prime number, then there is a prime
q
satisfying
p < q < p
+ 6.
(d) There exists a positive integer
m
such that 30
,
77
, m
are pairwise relatively
prime.
(e) In the RSA cryptosystem, when Bob wants to send Alice a message, it is
essential that Bob keeps his encryption method secret.
(f) 3
405
≡
13 (mod 7).
(g) It is valid to deduce
¬
q
, if
¬
p
and
p
→
q
have been proved already.
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 Spring '08
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 Math, Natural number, Prime number, Prof. Bjorn Poonen, encryption method secret

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