mathematical induction to show that
(cos x + i sin x)n = cos nx + i sin nx
whenever n is a positive integer. (Here
i is the square root of 1.) [Hint: Use
the identities cos(a + b) = cos a cos b
sin a sin b and sin(a + b) = sin a cos b +
cos a sin b.] 26.

lines in the plane you only need two
colors to color the regions formed so
that no two regions that have an edge
in common have a common color. 35.
Show that n! can be represented as the
sum of n of its distinct positive divisors
whenever n 3. [Hint: Use

rational number p/q with 0 < p/q < 1
as the sum of distinct unit fractions. At
each step of the algorithm, we find the
smallest positive integer n such that
1/n can be added to the sum without
exceeding p/q. For example, to express
5/7 we first start the

corresponds to a choice of one of the n
elements in the codomain for each of
the m elements in the domain. Hence,
by the product rule there are n n
n = nm functions from a set with m
elements to one with n elements. For
example, there are 53 = 125 differ

5. Show that 1 1 4 + 1 4 7 + 1 (3n
2)(3n + 1) = n 3n + 1 whenever n is a
positive integer. 6. Use mathematical
induction to show that 2n > n2 + n
whenever n is an integer greater than
4. 7. Use mathematical induction to
show that 2n > n3 whenever n is an

called strictly increasing if each term is
larger than the one that precedes it,
and it is called strictly decreasing if
each term is smaller than the one that
precedes it. THEOREM 3 Every
sequence of n2 + 1 distinct real
numbers contains a subsequence of

Under the new plan, there are 800
800 10,000 = 6,400,000,000 different
numbers available. EXAMPLE 9
What is the value of k after the
following code, where n1, n2,.,nm are
positive integers, has been executed?
k := 0 for i1 := 1 to n1 for i2 := 1 to n2

integer N with N/k > r 1, namely, N =
k(r 1) + 1, is the smallest integer
satisfying the inequality N/k r. Could
a smaller value of N suffice? The
answer is no, because if we had k(r 1)
objects, we could put r 1 of them in
each of the k boxes and no box w

recursively versus that needed to
compute them iteratively. Writing
Projects Respond to these with essays
using outside sources. 1. Describe the
origins of mathematical induction.
Who were the first people to use it and
to which problems did they apply it

Rosen-2311T MHIA017-Rosen-v5.cls
May 13, 2011 10:25 382 5 / Induction
and Recursion The set B of all balanced
strings of parentheses is defined
recursively by B, where is the
empty string; (x) B, xy B if x, y B.
59. Show that ( )( ) is a balanced
string o

among 25 million telephones, at least
25,000,000/8,000,000 = 4 of them
must have identical phone numbers.
Hence, at least four area codes are
required to ensure that all 10-digit
numbers are different. Example 9,
although not an application of the
general

14. a) Does testing a computer
program to see whether it produces
the correct output for certain input
values verify that the program always
produces the correct output? b) Does
showing that a computer program is
partially correct with respect to an
initi

Supplementary Exercises 381 47. Is
this proof that 1 1 2 + 1 2 3 + 1 (n
1)n = 3 2 1 n , whenever n is a
positive integer, correct? Justify your
answer. Basis step: The result is true
when n = 1 because 1 1 2 = 3 2 1 1 .
Inductive step: Assume that the re

every other city either directly or via
exactly one other city. 39. Use
mathematical induction to show that
when n circles divide the plane into
regions, these regions can be colored
with two different colors such that no
regions with a common boundary ar

rational. Show that the set of positive
integers of the form b 2 has a least
element a. Then show that a 2 a is a
smaller positive integer of this form.]
52. A set is well ordered if every
nonempty subset of this set has a least
element. Determine whether

is uniquely defined for all nonnegative
integers n. 72. Prove that a(n) = (n
+ 1) where = (1 + 5)/2. [Hint:
First show for all n > 0 that (n n) +
(2n 2n) = 1. Then show for all real
numbers with 0 < 1 and = 1
that (1 + )(1 ) + + = 1,
considering the case

were done, then there are n1 n2
nm ways to carry out the procedure.
This version of the product rule can be
proved by mathematical induction
from the product rule for two tasks
(see Exercise 72). EXAMPLE 4 How
many different bit strings of length
seven a

b), where a and b are nonnegative
integers not both zero, based on these
facts: gcd(a, b) = gcd(b, a) if a>b, gcd(0,
b) = b, gcd(a, b) = 2 gcd(a/2, b/2)if a
and b are even, gcd(a, b) = gcd(a/2, b)
if a is even and b is odd, and gcd(a, b) =
gcd(a, b a). 67

n or fewer symbols where each symbol
is T, F, one of the propositional
variables p and q, or an operator from
cfw_, , ,. 4. Given a string, find its
reversal. 5. Given a real number a and a
nonnegative integer n, find an using
recursion. 6. Given a real n

pigeonhole principle, the objects to be
placed in boxes must be chosen in a
clever way. A few such applications will
be described here. EXAMPLE 10 During
a month with 30 days, a baseball team
plays at least one game a day, but no
more than 45 games. Show

string. a) Find all strings in S of length
not exceeding five. b) Give an explicit
description of the elements of S. 58. Let
S be the set of strings defined
recursively by abc S, bac S, and acb
S, where a, b, and c are fixed letters;
and for all x S, abc

(x) = ex and g(x) = ecx , where c is a
constant. Use mathematical induction
together with the chain rule and the
fact that f (x) = ex to prove that g(n) =
cnecx whenever n is a positive integer.
19. Formulate a conjecture about
which Fibonacci numbers are

n j = 1 aj n j = 1 bj (mod m). b) n j = 1
aj n j = 1 bj (mod m). 15. Show that if
n is a positive integer, then
n k = 1 k + 4 k(k + 1)(k + 2) = n(3n + 7)
2(n + 1)(n + 2) . 16. For which positive
integers n is n + 6 < (n2 8n)/16? Prove your answer using ma

problems related to the reasoning and
information needs required for
intelligent computer behavior.
McCarthy was among the first
computer scientists to design timesharing computer systems. He
developed LISP, a programming
language for computing using symb

the arrangements of a specified kind.
This is often important in computer
simulations. We will devise algorithms
to generate arrangements of various
types. 6.1 The Basics of Counting
Introduction Suppose that a password
on a computer system consists of si

Rosen-2311T MHIA017-Rosen-v5.cls
May 13, 2011 10:25 6 CHAPTER
Counting 6.1 The Basics of Counting 6.2
The Pigeonhole Principle 6.3
Permutations and Combinations 6.4
Binomial Coefficients and Identities
6.5 Generalized Permutations and
Combinations 6.6 Gen

least one pair of these points has
integer coordinates. 12. How many
ordered pairs of integers (a, b) are
needed to guarantee that there are two
ordered pairs (a1, b1) and (a2, b2)
such that a1 mod 5 = a2 mod 5 and b1
mod 5 = b2 mod 5? 13. a) Show that if

of the North American Numbering
Plan.) As will be shown, the new plan
allows the use of more numbers. In the
old plan, the formats of the area code,
office code, and station code are NYX,
NNX, and XXXX, respectively, so that
telephone numbers had the form

integer N such that N/5 = 6. The
smallest such integer is N = 5 5 + 1 =
26. If you have only 25 students, it is
possible for there to be five who have
received each grade so that no six
students have received the same grade.
Thus, 26 is the minimum number

to choose a member of the
mathematics faculty and there are 83
ways to choose a student who is a
mathematics major. Choosing a
member of the mathematics faculty is
never the same as choosing a student
who is a mathematics major because
no one is P1: 1 CH0