Math 55. Sample Answers to First Midterm
There may be other correct solutions to these problems.
1.
(8 points) Use rules of inference to show that the hypotheses:
“Today is Valentine’s Day,”
“If today is Valentine’s Day then Fred buys chocolate on his way to campus (today),” and
“Every day, if Fred oversleeps then he does not buy chocolate on his way to campus”
imply the conclusion
“Fred did not oversleep today.”
Let
p
be “Today is Valentine’s Day,”, let
q
(
d
) be “Fred buys chocolate on day
d
,
and let
r
(
d
) be “Fred oversleeps on day
d
.” Then our argument is:
Step
Reason
1.
p
Premise
2.
p
→
q
(today)
Premise
3.
q
(today)
Modus Ponens using (1) and (2)
4.
¬¬
q
(today)
Double negation rule
5.
∀
d
(
r
(
d
)
→ ¬
q
(
d
))
Premise
6.
r
(today)
→ ¬
q
(today)
Universal instantiation of (4)
7.
¬
r
(today)
Modus Tollens using (4) and (6)
2.
(9 points) For each of the following functions, determine:
(i) whether it is 1

1 ,
(ii) whether it is onto, and
(iii) whether it has an inverse.
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 Spring '08
 STRAIN
 Math, Logic, right endpoint moves, premise premise modus, negation rule Premise, Fred oversleeps, sorted list c1

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