Chapter 2
The Logic of Quantified Statements
Section 2.1
Intro to Predicates & Quantified
Statements
Predicate Calculus
• The symbolic analysis of predicates and
quantified statements is known as predicate
calculus.
• Predicate calculus is used to determine the
validity of statements like:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
Predicate
• Predicate is the part of the sentence that
provides information about the subject.
– Example: “Dr. Ricanek is a resident of New
Hanover County”
• Subject: Dr. Ricanek
• Predicate: is a resident of New Hanover County
Predicate
• Predicate can be formed by removing the subject.
– Example: “Dr. Ricanek is a resident of New Hanover
County”
• predicate symbol P = “is a resident of New Hanover County”
• P(x) = x is a resident of New Hanover County
• x is predicate variable, when it is giving a concrete value P(x)
becomes a statement
• x = Karl Ricanek
• P(x) = “Karl Ricanek is a resident of New Hanover County”
Predicate
• Predicate can be formed by removing the
nouns.
– Example: “Dr. Ricanek is a resident of New
Hanover County”
•
•
•
•
•
predicate symbol Q = “is a resident of”
Q(x,y) = x is a resident of y
x,y are predicate variables
x = Charlotte, y = Pender County
Q(x,y) = “Charlotte is a resident of Pender County”
Predicate
• Definition:
– 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 of a predicate variable is the set of all
values that may be substituted in place of the
variable.
Example
• P = “is a public university in the UNC system”
• P(x) = x is a public university in the UNC
System.
• predicate variable x, domain is any one of the
16 universities in UNC system.
Example
• P(x) is the predicate “x2 > x”, domain of x is all
real numbers, R.
• Determine which is valid:
– P(2): 22 > 2
– P(1/2): ½2 > ½
– P(-1/2): -½2 > -½, ¼ > -½
Number Systems & Notations
• There are universally accepted symbols and
notations in mathematics, i.e.
– R - set of all real numbers
– Z - set of all integers
– Q - set of all rational numbers, quotients of
integers
– + - all positive numbers
– - - all negative numbers
• Example: R+, Z-
Number Systems & Notations
– denotes a member of
– x ∈A, x is a member of set A
– x ∉ A, x is not a member of set A
– … (ellipsis), “and so forth”
– | “such that”
– Example:
• { x ∈ D | P(x) }, “the set of all x in D such that P(x)”
Truth Set
• A truth set is the set of all elements of D that
make P(x) true when they are substituted for
x.
• Truth set is denoted: { x ∈ D | P(x) }
Example
• Let Q(n) be the predicate “n is a factor of 12.”
Find the truth set of Q(n) if
– a. the domain of n is the set of Z+ (positive
integers)
– solution: truth set is {1, 2, 3, 4, 6, 12}
• Let Q(n) be the predicate “n is a factor of 6.”
Find the truth set of Q(n) if
– a. the domain of n is the set Z (all integers)
– solution: truth set is {1, 2, 3, 6, -1. -2, -3, -6}
Universal Quantifier
• Universal quantifier symbol: ∀
• ∀denotes “for all”
– Example:
• “All human beings are mortal”
• ∀human beings x, x is mortal, or
• ∀x ∈S, x is mortal (What does S denote?)
Universal Statement
• A universal statement has the form, ∀x ∈ D,
Q(x).
• Universal statement is true, if and only if, all
Q(x) is true for every x in (domain).
• Counterexample occurs when a x∈D, Q(x) is
false.
Example
• Let D = {1, 2, 3, 4, 5}, and consider the
statement ∀x ∈ D, x2 ≥ x. Show that this
statement is true.
– 12 ≥ 1, 22 ≥ 2, 32 ≥ 3, 42 ≥ 4, 52 ≥ 5; Hence, true.
– Proof by exhaustion…
• Consider, ∀x ∈ R, x2 ≥ x
– find one case where not true (counterexample)
– x = ½ , ½ 2 ≥ ½; Hence, statement is false by
counterexample.
Existential Quantifier
• Existential quantifier symbol: ∃
• ∃denotes “there exists”.
– Example:
• “There is a student in CSC 133”
• ∃a person s such that s is a student in CSC 133, or
• ∃s ∈ S | s is a student in CSC 133
Existential Quantifier
• Existential statement has the form, ∃x ∈ D |
Q(x).
• Existential is defined to be true if, and only if,
Q(x) is true for at least one x in D.
• It is false if, and only if, Q(x) is false for all x in
D.
Example
• Consider, ∃m ∈ Z | m*m = m
– only have to find 1-case where this is true
– if m = 1, then 1*1 = 1; hence, statement true.
Universal Conditional Statement
• ∀x, if P(x) then Q(x)
• Example:
– ∀x ∈ R, if x > 2 then x2 > 4
– iinformal
• If a real number is greater than 2, then its square is
greater than 4, or
• The square of any real number that is greater than 2 is
greater than 4.
Example
• Formal and informal examples of universal
conditional statement
– ∀x ∈ R, if x ∈ Z then x ∈ Q
– ∀ real numbers x, if x is an integer, then x is a
rational number.
– “If a real number is an integer, then it is a rational
number.”
Equivalent Forms of ∀&∃
• There are equivalent forms of universal and
existential statements.
– Example: “All integers are rational.”
• ∀real numbers x, if x is an interger then x is rational
• ∀ integers x, x is rational
– ∀x ∈U, if P(x) then Q(x) ≡∀x ∈D, Q(x)
– ∃x such that P(x) and Q(x) ≡∃x ∈D such that Q(x)