P (xn), because this disjunction is
true if and only if at least one of P (x1), P
(x2), . . . , P (xn) is true. EXAMPLE 16 What
is the truth value of xP (x), where P (x) is
the statement x2 > 10 and the universe
of discourse consists of the positive
inte

14 Let P (x) denote the statement x > 3.
What is the truth value of the
quantification xP (x), where the domain
consists of all real numbers? Solution:
Because x > 3 is sometimes truefor
instance, when x = 4the existential
quantification of P (x), which i

consider the proposition There is a
student in this class who has taken a
course in calculus. This is the existential
quantification xQ(x), where Q(x) is the
statement x has taken a course in
calculus. The negation of this statement
is the proposition It

equivalent if and only if they have the
same truth value no matter which
predicates are substituted into these
statements and which domain of
discourse is used for the variables in
these propositional functions. We use the
notation S T to indicate that tw

q, each a bit, and produces as output the
signal p q. Finally, the AND gate takes
two input signals p and q, each a bit, and
produces as output the signal p q. We
use combinations of these three basic
gates to build more complicated circuits,
such as that

children. P1: 1/1 P2: 1/2 QC: 1/1 T1: 2
CH01-7T Rosen-2311T MHIA017-Rosenv5.cls May 13, 2011 15:27 20 1 / The
Foundations: Logic and Proofs EXAMPLE
8 A father tells his two children, a boy and
a girl, to play in their backyard without
getting dirty. Howev

1), the variable x is bound by the
existential quantification x, but the
variable y is free because it is not bound
by a quantifier and no value is assigned to
this variable. This illustrates that in the
statement x(x + y = 1), x is bound, but y
is free.

exists a class c such that professor p is the
instructor of class c and student s is
enrolled in class c. (Note that a comma is
used to represent a conjunction of
predicates in Prolog. Similarly, a
semicolon is used to represent a
disjunction of predicate

the statements There is an honest
politician and All Americans eat
cheeseburgers? Solution: Let H (x)
denote x is honest. Then the statement
There is an honest politician is
represented by xH (x), where the
domain consists of all politicians. The
negation

true, but does not know whether d is true.
After the son has answered No to the
first question, the daughter can determine
that d must be true. This follows because
when the first question is asked, the son
knows that s d is true, but cannot
determine whe

Translating sentences in English (or other
natural languages) into logical
expressions is a crucial task in
mathematics, logic programming,
artificial intelligence, software
engineering, and many other disciplines.
We began studying this topic in Section

Q(x) and xP (x) xQ(x) are logically
equivalent by doing two things. First, we
show that if x(P (x) Q(x) is true, then
xP (x) xQ(x) is true. Second, we
show that if xP (x) xQ(x) is true, then
x(P (x) Q(x) is true. So, suppose that
x(P (x) Q(x) is true. Thi

support the use of quotation marks to
search for specific phrases. So, it may be
more effective to search for pages
matching New Mexico AND
UNIVERSITIES.) P1: 1/1 P2: 1/2 QC: 1/1
T1: 2 CH01-7T Rosen-2311T MHIA017Rosen-v5.cls May 13, 2011 15:27 1.2
Applica

techniques from propositional logic, they
are called Boolean searches. In Boolean
searches, the connective AND is used to
match records that contain both of two
search terms, the connective OR is used to
match one or both of two search terms,
and the conn

philosophical essays and aphorisms, and
several advanced books on mathematical
logic and set theory. He is particularly
interested in self-reference and has
worked on extending some of Gdels
results that show that it is impossible to
write a computer prog

and q be the statements that A is a knight
and B is a knight, respectively, so that p
and q are the statements that A is a
knave and B is a knave, respectively. We
first consider the possibility that A is a
knight; this is the statement that p is true.
If

universal quantification, namely, xP (x),
where P (x) is the statement x has taken
a course in calculus and the domain
consists of the students in your class. The
negation of this statement is It is not the
case that every student in your class has
taken

paid the subscription fee and logical
connectives (including negations). a) The
user has paid the subscription fee, but
does not enter a valid password. b)
Access is granted whenever the user has
paid the subscription fee and enters a
valid password. c) A

UNIVERSITIES but do not contain the
word NEW. (In Google, and many other
search engines, the word NOT is
replaced by the symbol -. In Google, the
terms used for this last search would be
MEXICO UNIVERSITIES -NEW.) Logic
Puzzles Puzzles that can be solved

r, we use an inverter to produce r from
the input r. Then, we use an OR gate to
combine p and r. To build a circuit for p
(q r), we first use an inverter to
obtain r. Then we use an OR gate with
inputs q and r to obtain q r. Finally,
we use another inver

integers, we give a counterexample. We
see that x = 0 is a counterexample because
x2 = 0 when x = 0, so that x2 is not greater
than 0 when x = 0. Looking for
counterexamples to universally quantified
statements is an important activity in the
study of mat

quantification of P (x) is the statement P
(x) for all values of x in the domain. The
notation xP (x) denotes the universal
quantification of P (x). Here is called the
universal quantifier. We read xP (x) as
for all xP (x) or for every xP (x). An
element

SMULLYAN (BORN 1919) Raymond
Smullyan dropped out of high school. He
wanted to study what he was really
interested in and not standard high school
material. After jumping from one
university to the next, he earned an
undergraduate degree in mathematics at

QC: 1/1 T1: 2 CH01-7T Rosen-2311T
MHIA017-Rosen-v5.cls May 13, 2011
15:27 44 1 / The Foundations: Logic and
Proofs THE UNIQUENESS QUANTIFIER
We have now introduced universal and
existential quantifiers. These are the most
important quantifiers in mathemat

restriction of an existential quantification
is the same as the existential
quantification of a conjunction. For
instance, z > 0 (z2 = 2) is another way of
expressing z(z > 0 z2 = 2). Precedence
of Quantifiers The quantifiers and
have higher precedence t

besides calculus, we may prefer to use the
two-variable quantifier Q(x, y) for the
statement student x has studied subject
y. Then we would replace C(x) by Q(x,
calculus) in both approaches to obtain
xQ(x, calculus) or x(S(x) Q(x,
calculus). P1: 1/1 P2: 1

value of xP (x) depends on the domain!
Besides for all and for every, universal
quantification can be expressed in many
other ways, including all of, for each,
given any, for arbitrary, for each, and
for any. Remark: It is best to avoid using
for any x be

are dull in color. Hummingbirds are
small. Let P (x), Q(x), R(x), and S(x) be
the statements x is a hummingbird, x is
large, x lives on honey, and x is richly
colored, respectively. Assuming that the
domain consists of all birds, express the
statements in

statement so that we can clearly identify
the appropriate quantifiers to use. Doing
so, we obtain: For every student in this
class, that student has studied calculus.
Next, we introduce a variable x so that our
statement becomes For every student x
in thi