and let f (x) > 0 for all x R. Show that f
(x) is strictly increasing if and only if the
function g(x) = 1/f (x) is strictly
decreasing. 25. Let f : R R and let f (x) >
0 for all x R. Show that f (x) is strictly
decreasing if and only if the function g(x)
are added, the binomial coefficient in
the next row between these two
coefficients is produced. BLAISE
PASCAL (16231662) Blaise Pascal
exhibited his talents at an early age,
although his father, who had made
discoveries in analytic geometry, kept
mathemat
integer, then 2n n = n k = 0 n k 2 .
Proof: We use Vandermondes identity
with m = r = n to obtain 2n n = n k = 0
n n k n k = n k = 0 n k 2 . The last
equality was obtained using the
identity n k = n n k . ALEXANDRETHOPHILE VANDERMONDE (1735
1796) Because
integer greater than 1, then n n/2
2n/n. b) Conclude from part (a) that
if n is a positive integer, then 2n n
4n/2n. 17. Show that if n and k are
integers with 1 k n, then n k
nk/2k1. 18. Suppose that b is an
integer with b 7. Use the binomial
theorem
FUNCTIONS We will now describe an
important application of the concepts of
this section to computer science. In
particular, we will show that there are
functions whose values cannot be
computed by any computer program.
DEFINITION 4 We say that a function
columns. EXAMPLE 2 We have 1 0 1 2
2 3 34 0 + 3 4 1 1 3 0 112
= 4 4 2 3 1 3 252 . P1: 1
CH02-7T Rosen-2311T MHIA017-Rosenv5.cls May 13, 2011 10:24 2.6 Matrices
179 We now discuss matrix products.A
product of two matrices is defined only
when the number o
complexity of an algorithm and g(n) is a
reference function, means that C1g(n) f
(n) C2g(n) when n>k, where C1, C2, and
k are constants. So without knowing the
constants C1, C2, and k in the inequality,
this estimate cannot be used to determine
a lower bo
zeroone matrix with (i, j )th entry aij
bij . The join of A and B is denoted by A
B. The meet of A and B is the zeroone
matrix with (i, j )th entry aij bij . The
meet of A and B is denoted by A B.
EXAMPLE 7 Find the join and meet of the
zeroone matrices
seat these children in a row of chairs if
the identical triplets or twins cannot
be distinguished from one another? 20.
How many solutions are there to the
inequality x1 + x2 + x3 11, where x1,
x2, and x3 are nonnegative integers?
[Hint: Introduce an auxi
as a sum of terms involving binomial
coefficients. We will prove this
theorem using a combinatorial proof.
We will also show how combinatorial
proofs can be used to establish some of
the many different identities that
express relationships among binomial
NP problem asks whether NP, the class of
problems for which it is possible to check
solutions in polynomial time, equals P, the
class of tractable problems. If P=NP, there
would be some problems that cannot be
solved in polynomial time, but whose
solution
STEPHEN COOK (BORN 1939) Stephen
Cook was born in Buffalo where his father
worked as an industrial chemist and
taught university courses. His mother
taught English courses in a community
college. While in high school Cook
developed an interest in electron
simultaneously. Efficient algorithms,
including most algorithms with
polynomial time complexity, benefit most
from significant technology
improvements. However, these
technology improvements TABLE 2 The
Computer Time Used by Algorithms.
Problem Size Bit O
strings BCA and ABF? 23. How many
ways are there for eight men and five
women to stand in a line so that no two
women stand next to each other? [Hint:
First position the men and then
consider possible positions for the
women.] 24. How many ways are there
that a finite group of guests arriving at
Hilberts fully occupied Grand Hotel can
be given rooms without evicting any
current guest. 6. Suppose that Hilberts
Grand Hotel is fully occupied, but the
hotel closes all the even numbered rooms
for maintenance.
position are equal. EXAMPLE 1 The
matrix 1 1 0 2 1 3 is a 3 2 matrix.
We now introduce some terminology
about matrices. Boldface uppercase
letters will be used to represent matrices.
DEFINITION 2 Let m and n be positive
integers and let A = a11 a12 .
a1n
positive integer, then there is a one-to-one
correspondence between S and T . 80.
Show that a set S is infinite if and only if
there is a proper subset A of S such that
there is a one-to-one correspondence
between A and S. P1: 1 CH02-7T Rosen2311T MHIA017
columns of A. In other words, if At = [bij ],
then bij = aj i for i = 1, 2,.,n and j = 1,
2,.,m. EXAMPLE 5 The transpose of the
matrix 123 456 is the matrix 1 4 2 5 3
6 . Matrices that do not change
when their rows and columns are
interchanged are often i
makes it true, but no polynomial time
algorithm has been discovered for finding
such an assignment of truth values. (For
example, an exhaustive search of all
possible truth values requires (2n) bit
operations where n is the number of
variables in the comp
rational number as a string of digits with
a slash and possibly a minus sign.] 27.
Show that the union of a countable
number of countable sets is countable. 28.
Show that the set Z+ Z+ is countable.
29. Show that the set of all finite bit
strings is count
function from the set A to the set B. Let S
be a subset of B. We define the inverse
image of S to be the subset of A whose
elements are precisely all pre-images of
all elements of S. We denote the inverse
image of S by f 1(S), so f 1(S) = cfw_a A |
f (a)
|A| |Z+|. 14. Show that if A and B are
sets with the same cardinality, then |A|
B| and |B|A|. 15. Show that if A and B
are sets, A is uncountable, and A B, then
B is uncountable. 16. Show that a subset
of a countable set is also countable. 17. If
A is an
an x must be chosen in two of the three
sums (and consequently a y in the
other sum). Hence, the number of such
terms is the number of 2-combinations
of three objects, namely, 3 2 . Similarly,
the number of terms of the form xy2 is
the number of ways to p
The continuum hypothesis was stated by
Cantor in 1877. He labored unsuccessfully
to prove it, becoming extremely dismayed
that he could not. By 1900, settling the
continuum hypothesis was considered to
be among the most important unsolved
problems in math
specified. If the score is still tied at the
end of the 10 penalty kicks, this
procedure is repeated. If the score is
still tied after 20 penalty kicks, a
sudden-death shootout occurs, with
the first team scoring an unanswered
goal victorious. a) How many
if a) S = cfw_2, 1, 0, 1, 2, 3. b) S = cfw_0, 1, 2, 3,
4, 5. c) S = cfw_1, 5, 7, 11. d) S = cfw_2, 6, 10,
14. 32. Let f (x) = 2x where the domain is
the set of real numbers. What is a) f (Z)?
b) f (N)? c) f (R)? 33. Suppose that g is a
function from A to
the terms of a sequence specified via a
recurrence relation. Identifying a
sequence when the first few terms are
provided is a useful skill when solving
problems in discrete mathematics. We
will provide some tips, including a useful
tool on the Web, for d
questions on a discrete mathematics
final exam. How many ways are there
to assign scores to the problems if the
sum of the scores is 100 and each
question is worth at least 5 points? 28.
Show that there are C(n + r q1 q2
qr 1, n q1 q2 qr)
different unord
the set of positive integers as an infinite
bit string with ith bit 1 if i belongs to the
subset and 0 otherwise. Suppose that you
can list these infinite strings in a sequence
indexed by the positive integers.
Construct a new bit string with its ith bit