Quantifiers and Negation

1. Quantifiers and Negation Predicates

2. Predicate • Will define function. • The domain of a function is the set from which possible values may be chosen for the argument of the function. • A predicate is a function whose possible values are True and False.

3. Universal Quantifier •  The universal quantifier. • Upside down A. • “For All”

4. Universal Quantifier • Let P(x) be a predicate with domain D. • A universal statement is a statement in the form “x, P(x)”. It is true iff P(x) is true for all x in D. It is false iff P(x) is false for at least one x in D. • A value of x which shows that P(x) is false is called a counterexample to the universal statement.

5. Universal Quantifier Examples • Let D = {1, 2, 3, 4, 5}. x, x² >= x • Let D=Reals. x  Reals, x² >= x • Let D= Natural numbers. Let P(x)=x is prime, E(x)=x is even. x E(x) x P(x) x (E(x) V E(x+1))

6. Existential Quantifier •  Existential quantifier . • Backwards E. • There Exists

7. Existential Quantifier • Let P(x) be a predicate with domain D. • An existential statement is a statement in the form “x, P(x)”. It is true iff P(x) is true for at least one x in D. It is false iff P(x) is false for all x in D. • Examples: D=Integers, m, m² = m D = {5, 6, 7, 8, 9}, m, m² = m

8. Universal Conditional Statement • Universal conditional statement x, (P(x)  Q(x)) Example: D=Reals.x, (x > 2  x2 > 4) • A universal conditional statement is vacuously true or true by default iff P(x) is false for every x in D. i.e. ~ xD, P(x) iff x D (P(x)  Q(x)) is vacuously true.

9. Universal Conditional Statement Examples: D=Natural numbers. E(x)=x is even. P(x)=x is prime x (E(x)  ~E(x+1)) x ((E(x) Λx > 2)  ~P(x)) x (P(x)  ~E(x))

10. Negation of Quantified Statements • ~ (xD, P(x)) ≡x  D, ~P(x) • ~(x  D, P(x))≡x  D, ~P(x) • ~ x  D, P(x)  Q(x) ≡ x  D, ~(P(x)  Q(x)) ≡ x  D, P(x)  ~Q(x)

11. Multiply Quantified Statements • Let the D=Integers. Let P(x, y) = y<x. • There exists a number x, there exists a number y, such that y<x x, y, P(x, y) ≡ y, x, P(x, y) True (both True) • For all numbers x, for all numbers y, y < x x, y, P(x, y) ≡ y, x, P(x, y) True (both False)

12. Multiply Quantified Statements • For all numbers x, there exists a number y such that x < y x, y, P(x, y) True • There exists a number y such that for all numbers x, x < y y, x, P(x, y) False • x, y, P(x, y) ≡ y, x, P(x, y) False

13. Negation of Multiply Quantified Statements • ~ x, y, P(x, y) ≡ x, y, ~P(x, y) • ~ x, y, P(x, y) ≡ x, y,~P(x, y) • ~ x, y, P(x, y) ≡x, y, ~P(x, y) • ~ x, y, P(x, y)≡x, y, ~P(x, y)

14. Example a) Symbolize: The sum of any two even integers is even. b) Negate it.

15. Example Solution • Let E(x)=“x is even”. • Let E(x, y)=“x+y is even”. • Let the domain be the Integers. a) x, y, E(x)^E(y) E(x, y) b)  x, y, E(x) ^ E(y) ^ ~E(x, y)

16. Which of the following is true? x  D, (P(x)  Q(x)) ≡ (x  D, P(x))  (x  D, Q(x)) x  D, (P(x)  Q(x)) ≡ (x  D, P(x))  (x  D, Q(x)) x  D, (P(x)  Q(x)) ≡ (x  D, P(x))  (x  D, Q(x)) x  D, (P(x)  Q(x)) ≡ (x  D, P(x))  (x  D, Q(x))

17. , , Λ, ν Let P(x) be a predicate. Let D={x1, x2, …, xn} be the domain of x. • xD, P(x)≡P(x1) ΛP(x2)Λ…ΛP(xn) • xD, P(x) ≡P(x1) νP(x2)ν…νP(xn)