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
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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..
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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)?
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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.
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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 in
D. A value for x for which Q(x) is false is called a
counterexample to the universal statement.
• 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.
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Formal Versus Informal
• When working with mathematics, it is
important to be able to translate between
formal (symbols) to informal (words) and visa
versa.
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Example – pg. 107 # 15
• Rewrite the following statements informally in
at least two different ways without using
variables or quantifiers.
a.  rectangles x, x is a quadrilateral.
b.  a set A such that A has 16 subsets.
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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 be the set D of objects discussed in mathematics
courses,
• let Real(x) be “x is a real number”,
• let Pos(x) be “x 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.
b.
c.
d.
Pos(0)
x, Real(x)  Neg(x)  Pos(-x).
x, Int(x)  Real(x).
 x s.t. Real (x)  ~Int (x).
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