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, x∀ ∈ 2 ≥ 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)∃ ∈