MAD2104/Final Exam Page 8 of 8
17. (10 pts.) The following is a valid argument:
"Each of five friends, Larry, Moe, Curly, Samantha, and
Anitra, has taken a course in discrete mathematics and passed
with a grade of at least a C. Every student who has taken and
passed a discrete math class with a grade of at least a C can
take a course in algorithms. Therefore, all five friends can
take a course in algorithms."
The validity of the argument can be seen easily by
symbolizing the argument using propositional functions and
quantifiers as follows:
Define propositional functions as follows:
f(x) : "x is one of the friends listed."
d(x) : "x has taken and passed discrete math with at least a C."
a(x) : "x can take a course in algorithms."
Then the argument translates into this:
(
∀
x)(f(x)
→
d(x))
(
∀
x)(d(x)
→
a(x))
∴
(
∀
x)(f(x)
→
a(x))
In the proof of validity which follows provide the
justification(s) for each step. In doing this, you should
explicitly cite rules of inference and quantification,
hypotheses, and preceding steps by number.
1.
(
∀
x)(f(x)
→
d(x))
Justification:
2.
f(y)
→
d(y)
Justification:
3.
(
∀
x)(d(x)
→
a(x))
Justification:
4.
d(y)
→
a(y)
Justification:
5.
f(y)
→
a(y)
Justification:
6.
(
∀
x)(f(x)
→
a(x))
Justification:
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 Spring '08
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 Logic, pts, Natural number, MAD2104/Final Exam

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