calculus book - Mathematics and Computer Science

1.4. CALCULUS AND THE “REAL WORLD” 31
Notice that “for every” and “there exists” are exchanged under negation.
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Exercise 1.8 A game-show host presents the contestant with the equation
“a 2 + b 2 = c 2 .” The contestant replies, “What is the Pythagorean
Theorem?”
(a) Is this really the correct question? If not, can you append a clause
to give a question that would satisfy a mathematician?
(b) State the Pythagorean Theorem in if-then form.
A theorem is not merely its conclusion. Despite this, after A. Wiles
announced the proof of Fermat’s Last Theorem, the Mathematical Sciences
Research Institute (MSRI) in Berkeley produced T-shirts bearing
the message, “a n +b n = c n : NOT!” ⋄
Exercise 1.9 The President, a law-abiding citizen who always tells
the truth, has time for one more Yes/No question at a press conference.
In an attempt to embarrass the President, a reporter asks, “Have you
stopped offering illegal drugs to visiting Heads of State?”
(a) Which answer (“Yes” or “No”) is logically truthful?
(b) Suppose the President answers “Yes”. Can the public conclude
that the President has offered illegal drugs to visiting Heads of
State? What if the answer is “No”?
(c) Explain why both answers are embarrassing.
If the President were a Zen Buddhist, she might reply, “mu,” 5 meaning
“Your question is too flawed in its hypotheses to answer meaningfully.”
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Exercise 1.10 One striking peculiarity of the human brain is that it
is “better” at seeing certain situations in an emotional light than it is at
understanding an equivalent logical formulation. Here is an example.
(a) Each card in a deck is printed with a letter on one side and a
number on the other. Precisely, the letter is either “D” or “N”
and the number is a whole number between 16 and 70 inclusive.
There is no restriction on the combinations in which cards are
printed. Your job is to assess whether or not cards satisfy the
5 Pronounced “moo”