MAD 2104 Midterm: MAD 2104 FIU Exam 115k
MAD 2104 Jan 29, 2015
Exam I and Key Prof. S. Hudson
1) Write out (in words) the converse of this sentence:
If it is wet in India, then it is dry in Pakistan.
2) Give the precise definition of p→q(of ”if pthen q”). I am not looking for a wordy
discussion. Perhaps some kind of formula or chart would be better.
3) Compute (eg simplify) this formula: (0101 ∧0110) ∨0010.
4) [10pts each] Prove or disprove. That means - answer each of these with True or False.
Then explain and justify your reasoning. You can use well-known facts such as π < 4,
the transitive property, or anything proved in class. You can assume the universal set is
U=R, the real numbers. Be very careful to get the TF parts correct, otherwise I cannot
give partial credit. But your proofs will count at least as much.
4a) ∀x, x > π →x > 4.
4b) ∀x∈Q, x2= 2 →x < 10.
4c) ∀ǫ > 0,∃δ > 0, ǫδ = 2.
4d) ¬(p⊕q)≡(p→q)
5) Steve would like to determine the relative salaries of three coworkers using two facts.
First, he knows that if Fred is not the highest paid of the three, then Janice is. Second,
he knows that if Janice is not the lowest paid, then Maggie is paid the most. Is it possible
to determine the relative salaries of Fred, Maggie and Janice from what Steve knows ? If
so, who is paid the most and who the least ? Explain your reasoning (for max credit, this
should be a logical sequence of deductions, with clear reasons).
6) What rule of inference is used in this famous argument ? ”All men are mortal. Socrates
is a man. Therefore, Socrates is mortal.” If you do not know the name of the rule, you
can write it out instead (in the usual column format) for full credit.
7) [10pts total] Consider the proposition ∀x, (x > 2→x > 3). It is true for some domains
but not all.
7a) Give an example of a domain (a set of numbers, of course) for which it is false, with a
brief explanation.
7b) Give a domain for which it is true. No explanation required.
Remarks and Answers: The average among the top 35 grades was 77 out of 100, which
is very good, but not unusual for the first test. The 4 highest scores were all in the 91-92
range. The best results, 90% or better, were on problems 2, 3, 4a and 4d. The lowest
results, 50% to 60%, were on problems 4b, 4c and 6.
You can use the scale on the syllabus to convert your numerical score on this exam into a
letter grade. An 85, for example, is the low A-. It is fairly likely that the average on the
next exams will be lower than 77 and that I will lower the scale for those exams.
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Document Summary
Prof. s. hudson: write out (in words) the converse of this sentence: If it is wet in india, then it is dry in pakistan: give the precise de nition of p q (of if p then q ). I am not looking for a wordy discussion. Perhaps some kind of formula or chart would be better: compute (eg simplify) this formula: (0101 0110) 0010, [10pts each] prove or disprove. That means - answer each of these with true or false. You can use well-known facts such as < 4, the transitive property, or anything proved in class. Be very careful to get the tf parts correct, otherwise i cannot give partial credit. But your proofs will count at least as much. 4a) x, x > x > 4. 4b) x q, x2 = 2 x < 10. 4d) (p q) (p q: steve would like to determine the relative salaries of three coworkers using two facts.
