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

124 The Nuts and Bolts of Proof, Third Edition
SOLUTIONS FOR THE EXERCISES AT THE END OF
THE SECTIONS
BASIC TECHNIQUES TO PROVE IF/THEN STATEMENTS
1. There exists at least one real number for which the function/is not
defined. (Or: The function/is not defined for all real numbers.)
2. Let X and y be two numbers. There is no rational number z such that
x-\-z = y. (Or: Let x and y be two numbers. Then x-\-z^y for all
rational numbers z.)
3. The function / does not have the property that for any two distinct
real numbers x and y,fix)^f{y). (Or: There exist at least two distinct
numbers x and y for which/(x) =/(};).)
4. The equation P{x) — 0 has at least two solutions. (Or: The equation
P(x) = 0 has more than one solution.)
5. There is at least one nonzero real number that does not have a
nonzero opposite.
6. Either: (i) There exists a number n > 0 for which there is no number
M„ > 0 such that f(x) > n for all numbers x with x > M„; or (ii) there
exists a number n> 0 such that for every M„ > 0 there is at least one x
with x>Mn and/(x) < n.
7. There exists at least one number satisfying the equation P(x) = Q(x)
such that \x\> 5.
8. Compare this statement with statement 4. In this case, we do not
know whether a solution exists at all. So the answer is: Either the
equation P(x) = 0 has no solution or it has at least two solutions.
9. The function/is not continuous at the point c if there exists an ^ > 0
such that for every 8> 0 there exists an x with |x —c| < 8 and
\f{x)-f{c)\>8,
10. There exists at least one real number XQ such that/(xo) is an irrational
number. (Or: The function^x) is not rational for every real number x.)
11. (a) If X is an integer not divisible by 2, then x is not divisible by 6.
(b) If X is an integer divisible by 2, then x is divisible by 6. (c) If x
is an integer not divisible by 6, then x is not divisible by 2.
12. (a) If the diagonals of a quadrilateral bisect, then the quadrilateral is
a parallelogram, (b) If the diagonals of a quadrilateral do not bisect,
then the quadrilateral is not a parallelogram, (c) If a quadrilateral is a
parallelogram, then its diagonals bisect.