different executive positions on a
committee? 9. a) What is Pascals
triangle? b) How can a row of Pascals
triangle be produced from the one
above it? 10. What is meant by a
combinatorial proof of an identity?
How is such a proof different from an
algebrai

variables or negations of these variables.
We can replace a statement in
propositional logic that is not a clause by
one or more equivalent statements that
are clauses. For example, suppose we
have a statement of the form p (q r).
Because p (q r) (p q) (p

obtain by substituting these values of p
and q into the argument form is If you
have access to the network, then you can
change your grade. You have access to
the network. You can change your
grade. The argument we obtained is a
valid argument, but becaus

basis for system development. Example 3
shows how compound propositions can
be used in this process. EXAMPLE 3
Express the specification The automated
reply cannot be sent when the file system
is full using logical connectives. Solution:
One way to transl

TUKEY (19152000) Tukey, born in New
Bedford, Massachusetts, was an only
child. His parents, both teachers, decided
home schooling would best develop his
potential. His formal education began at
Brown University, where he studied
mathematics and chemistry.

these compound propositions. a) (p q)
(p q) b) (p q) (p q) c) (p q)
(p q) d) (p q) (p q) e) (p q)
(p r) f ) (p q) (p q) 34.
Construct a truth table for each of these
compound propositions. a) p p b) p
p c) p q d) p q e) (p q) (p
q) f ) (p q) (p q) 35

q) q) p is not a tautology, because it
is false when p is false and q is true.
However, there are many incorrect
arguments that treat this as a tautology. In
other words, they treat the argument with
premises p q and q and conclusion p as
a valid argumen

almost all the content of the text. Students
needing extra help will find tools on the
companion website for bringing their
mathematical maturity up to the level of
the text. The few places in the book where
calculus is referred to are explicitly noted.
M

1/1 T1: 2 CH01-7T Rosen-2311T
MHIA017-Rosen-v5.cls May 13, 2011
15:27 1.2 Applications of Propositional
Logic 17 often ambiguous. Translating
sentences into compound statements
(and other types of logical expressions,
which we will introduce later in this

specifications are consistent, we first
express them using logical expressions.
Let p denote The diagnostic message is
stored in the buffer and let q denote The
diagnostic message is retransmitted. The
specifications can then be written as p
q, p, and p

Hilbert, Ore, and Tao have been added.
Historical information has been added
throughout the text. Numerous updates
for latest discoveries have been made.
Expanded Media Extensive effort has been
devoted to producing valuable web
resources for this book. E

information. DEFINITION 7 A bit string is
a sequence of zero or more bits. The
length of this string is the number of bits
in the string. EXAMPLE 12 101010011 is
a bit string of length nine. We can
extend bit operations to bit strings. We
define the bitwi

will use propositional variables to
represent each sentence part and
determine the appropriate logical
connectives between them. In particular,
we let a, c, and f represent You can access
the Internet from campus, You are a
computer science major, and You

to 100 email addresses it finds in the
electronic message mailbox on this
personal computer. What is the
maximum number of different
computers this one computer can
infect in the time it takes for the
infected message to be forwarded five
times? 21. How m

algorithm for generating all the
combinations of the set of the n
smallest positive integers.
Supplementary Exercises 1. How many
ways are there to choose 6 items from
10 distinct items when a) the items in
the choices are ordered and repetition
is not al

elements? d) How many one-to-one
functions are there from a set with five
elements to a set with 10 elements? e)
How many onto functions are there
from a set with five elements to a set
with 10 elements? 4. How can you find
the number of possible outcomes

dollars or euros. 20. For each of these
sentences, determine whether an
inclusive or, or an exclusive or, is intended.
Explain your answer. a) Experience with
C+ or Java is required. b) Lunch includes
soup or salad. c) To enter the country you
need a pass

the most important rules of inference in
propositional logic. The tautology (p (p
q) q is the basis of the rule of
inference called modus ponens, or the law
of detachment. (Modus ponens is Latin for
mode that affirms.) This tautology leads
to the followi

setting up a correspondence between
the subsets of a set with an even
number of elements and the subsets of
this set with an odd number of
elements. [Hint:Take an element a in
the set. Set up the correspondence by
putting a in the subset if it is not
alre

p, q, and r and logical connectives
(including negations). P1: 1/1 P2: 1/2 QC:
1/1 T1: 2 CH01-7T Rosen-2311T
MHIA017-Rosen-v5.cls May 13, 2011
15:27 14 1 / The Foundations: Logic and
Proofs a) You get an A in this class, but
you do not do every exercise i

it is sunny, If we do not go swimming,
then we will take a canoe trip, and If we
take a canoe trip, then we will be home by
sunset lead to the conclusion We will be
home by sunset. Solution: Let p be the
proposition It is sunny this afternoon, q
the propo

argument form is a rule of inference from
Table 1. P1: 1/1 P2: 1/2 QC: 1/1 T1: 2
CH01-7T Rosen-2311T MHIA017-Rosenv5.cls May 13, 2011 15:27 72 1 / The
Foundations: Logic and Proofs TABLE 1
Rules of Inference. Rule of Inference
Tautology Name p p q q (p (p

the conclusion is true if the premises are
all true. From the definition of a valid
argument form we see that the argument
form with premises p1, p2,.,pn and
conclusion q is valid, when (p1 p2
pn) q is a tautology. The key to showing
that an argument in

Miami is the capital of Florida. c) 2 + 3 = 5.
d) 5 + 7 = 10. e) x + 2 = 11. f ) Answer this
question. 2. Which of these are
propositions?What are the truth values of
those that are propositions? a) Do not
pass go. b) What time is it? c) There are
no blac

containing at least N/k objects. P (n, r)
= n! (n r)! C(n, r) = n r = n! r!(n r)!
Pascals identity: n+1 k = n k1 + n k
the binomial theorem: (x + y)n = n k =
0 n k xnkyk There are nr rpermutations of a set with n elements
when repetition is allowed. There

Logic and Proofs EXAMPLE 7 Show that
the premises If you send me an e-mail
message, then I will finish writing the
program, If you do not send me an email message, then I will go to sleep
early, and If I go to sleep early, then I will
wake up feeling refr

e) You can access the website only if you
pay a subscription fee. f ) Getting elected
follows from knowing the right people. g)
Carol gets seasick whenever she is on a
boat. 23. Write each of these statements
in the form if p, then q in English. [Hint:
Re

if there must be at least two objects of
the first type, but no more than three
objects of the second type? 14. a) Let n
and r be positive integers. Explain why
the number of solutions of the
equation x1 + x2 + xn = r, where xi is
a nonnegative integer fo

p This argument uses the simplification
rule. P1: 1/1 P2: 1/2 QC: 1/1 T1: 2
CH01-7T Rosen-2311T MHIA017-Rosenv5.cls May 13, 2011 15:27 1.6 Rules of
Inference 73 EXAMPLE 5 State which rule
of inference is used in the argument: If it
rains today, then we wi

is best known for his invention, with J. W.
Cooley, of the fast Fourier transform. In
addition to his contributions to statistics,
Tukey was noted as a skilled wordsmith;
he is credited with coining the terms bit
and software. Tukey contributed his
insigh