Understanding Predicates and Quantified Statements in Predicate Calculus, Slides of Discrete Mathematics

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Chapter 2

The Logic of Quantified Statements

Section 2.

Intro to Predicates & Quantified Statements

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

• 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.

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)”

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?)

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, 2^2 ≥ 2, 3^2 ≥ 3, 4^2 ≥ 4, 5^2 ≥ 5; Hence, true.
• Proof by exhaustion…
• Consider, ∀ x ∈ R , x 2 ≥ 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

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 x 2 > 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.