1. Chapter 2 The Logic of Quantified Statements

2. Section 2.1 Intro to Predicates & Quantified Statements

3. 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.

4. 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

5. 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”

6. 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”

7. 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.

8. 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.

9. Example P(x) is the predicate “x 2 > x”, domain of x is all real numbers, R. Determine which is valid: – P(2): 2 2 > 2 – P(1/2): ½ 2 > ½ – P(-1/2): -½ 2 > -½, ¼ > -½

10. 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 -

11. 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)”

12. 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) }

13. 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}

14. 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?)

15. 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.

16. Example Let D = {1, 2, 3, 4, 5}, and consider the statement ∀ x ∈ D, x 2 ≥ x. Show that this statement is true. – 1 2 ≥ 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.

17. 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

18. 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.

19. 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.

20. 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.

21. 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.”

22. 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)