epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Exercises Without Solutions
Discuss the truth of the following statements. Prove the ones that are true;
find a counterexample for each one of the false statements. Exercises with the
symbol (*) require knowledge of calculus or hnear algebra.
GENERAL PROBLEMS
1. (I) Write each of the following statements in the form "If A, then B". (II)
Construct the contrapositive of each statement.
(a) Every differentiable function is continuous.
(b) The sum of two consecutive numbers is always an odd number.
(c) The product of two consecutive numbers is always an even number.
(d) No integer of the form n^ -f 1 is a multiple of 7.
(e) Two parabolas having three points in common coincide.
(f) Let b and c be any two real numbers with b<c, and let a be their
arithmetic average, defined as a = ^. Then b<a<c.
2. The number V7 is irrational.
3. The only prime of the form n^ — 1 is 31.
4. There is a differentiable function whose graph passes through the three
points (-1,0), (0, -3), and (1,5). (*)
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