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

112 The Nuts and Bolts of Proofs
65. Let |a„ = 4rf[ • Then lim a„ = 1.
66. Let c be a perfect square {i.e., c = a^ for some integer number a). Then
the number of distinct divisors of c is odd.
67. Let a, ft, c, and d be natural numbers such that ft is a multiple of a and d
is a multiple of c. Then ftd is a multiple of ac.
68. Let AuAi, ---.An be any n sets. Then (y4i U .42 U ... U Ar^'=
yl'i n ^2 (^ • • • ^ ^n for all n > 2.
69. There exists a third-degree polynomial whose graph passes through the
points(0,l), (-1,3), and(l,3).
70. The Fibonacci sequence /i,/2,...,/« is defined recursively as
/i = l,/2 = 1, and/„ =fn-\ +/„-2 for n > 2. Then for n > 2:
(a)/l+/2 + ....+/n=/n+2-l
(Hint.- Write several terms of the sequence to study its behavior.)
71. There exists a positive integer n such that n!<3".
72. Let a and ft be two real numbers with a <b. Then there exists a unique
number c, with a < c <b such that |a — c| = ^^-^
73. Let/(x) = |x|. Then lim/(x) = 0.
74. Let a and ft be two rational numbers, with a < b. Then there exist at least
four rational numbers between a and ft.
75. The sum of two increasing functions is an increasing function.
76. The sum of two prime numbers is not a prime number.
77. There is a digit that appears infinitely often in the decimal expansion
ofV2.
78. Let A and B be subsets of a universal set U. The symmetric difference of
A and B, indicated as A 05, is the subset of U defined as follows:
A^B = {A-B)U(B-A)
3
Show A^B on a Venn diagram.
Prove that (A-B)n(B-A) = 0.
Prove that A® B = (AU B) - (An B).
(You can use a Venn diagram as an example, but you need to write a
general proof as well.)
Prove that A®B = B eA.