Transcript 3.1 Predicates and Quantified Statements I
Discrete Structures
Chapter 3: The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I
… it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics. – A. N. Whitehead, 1861 – 1947
3.1 Predicates and Quantified Statements I 1
Definitions
• Predicate – A
predicate
is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables.
• Domain – The
domain
of a predicate variable is the set of all values that may be substituted in place of the variable.
• Premises – All statements in an argument and all statement forms in an argument form are called
premises
except for the last one..
3.1 Predicates and Quantified Statements I 2
Example – pg. 106 # 3
• Let
P
(
x
) be the predicate “
x
> 1/
x
”.
a. Write
P
(2),
P
(1/2),
P
(-1),
P
(-1/2), and
P
(-8), and indicate which of these statements are true and which are false.
, b. Find the truth set of
P
(
x
) if the domain of
x
is the set of all real numbers.
c. If the domain is the set + of all positive real numbers, what is the truth set of
P
(
x
)?
3.1 Predicates and Quantified Statements I 3
Quantifiers
We can obtain statements from predicates by adding quantifiers or words that refer to quantities.
• • The symbol denotes “for all” and is called the
universal quantifier
.
The symbol denotes “there exists” and is called the
existential quantifier
.
3.1 Predicates and Quantified Statements I 4
Definitions
• • Let
Q
(
x
) be a predicate and
D
the domain of
x
: Universal Statement – A
universal statement
is a statement of the form “
x
D
,
Q
(
x
).” It is defined to be true iff
Q
(
x
) is true for all
x
in
D
. It is defined to be false iff
Q
(
x
) is false for at least one
x D
. A value for
x
for which
Q
(
x
) is false is called a counterexample to the universal statement.
in Existential Statement – A
existential statement
is a statement of the form “
x
D
such that
Q
(
x
).” It is defined to be true iff
Q
(
x
) is true for at least one
x
in
D
. It is false iff
Q
(
x
) is false for all
x
in
D
. 3.1 Predicates and Quantified Statements I 5
Formal Versus Informal
• When working with mathematics, it is important to be able to translate between formal (symbols) to informal (words) and visa versa.
3.1 Predicates and Quantified Statements I 6
Example – pg. 107 # 15
• Rewrite the following statements informally in at least two different ways without using variables or quantifiers.
a.
b.
rectangles a set
A x
,
x
is a quadrilateral.
such that
A
has 16 subsets.
3.1 Predicates and Quantified Statements I 7
Example – pg. 108 #28
Rewrite each statement without using quantifiers or variables. Indicate which are true and which are false, and justify your answers as best as you can.
• • • • • Let the domain of
x
courses, be the set
D
of objects discussed in mathematics let Real(
x
) be “
x
let Pos(
x
) be “
x
is a real number”, is a positive real number”, let Neg(
x
) be “
x
is a negative real number”, and let Int(
x
) be “
x
is an integer.” a. Pos(0) b.
x
, Real(
x
) c.
d.
x
, Int(
x
)
x
s.t. Real (
x
Neg( )
x
) Real(
x
).
~Int (
x
Pos( ).
x
).
3.1 Predicates and Quantified Statements I 8