2. Predicates and Quantified Statements M260 3.1, 3.2

3. Predicate Example James is a student at Southwestern College. P(x,y) x is a student at y.

4. Predicate Truth Set P(x) is a predicate and x has domain D. The truth set of P(x): {x  D  P(x)}

5.  For all For every For arbitrary For any For each Given any

6. Universal Statement  x  D, Q(x) When true When false Counter examples

7.  There exists We can find a There is at least one For some For at least one

8. Existential Statement  x  D such that Q(x) When true When false

9. Translation Examples  x , x 2  0  x , x 2  -1  m  such that m 2 =m

10. Translation Examples If a real number is an integer then it is a rational number All bytes have eight bits No fire trucks are green.

11. Formal and Informal For all polygons p, if p is a square, then p is a rectangle. For all squares p, p is a rectangle. There exists a number n such that n is prime and n is even. There exists a prime number n such that n is even.

12. Trailing Quantifier For all squares p, p is a rectangle. p is a rectangle for any square p. There exists a prime number n such that n is even. n is even for some prime number n.

13. Equivalent forms of  and   x  U if P(x) then Q(x)  x  D, Q(x) Where D is all x such that P(x)

14. Implicit Quantification If n is a number, then it is a rational number If x>2 then x 2 >4. x>2  x 2 >4

15.  and  P(x)  Q(x) means that the truth set of P(x) is contained in the truth set of Q(x). P(x)  Q(x) means P(x) and Q(x) have identical truth sets.

16. Negation of Quantified Statements For all x in D, Q(x) There exists x in D such that ~Q(x). “all are” versus “some are not”

17. Negation of Quantified Statements There exists x in D such that Q(x) For all x in D, ~Q(x). “some are” versus “all are not”

18. Try These ~(For every prime p, p is odd) ~(There exists a triangle T, such that the sum of the angles of T equals 200 degrees)

19. No Politicians are Honest Formal Formal negation Informal negation

20. All computer programs are finite. Formal Formal negation Informal negation

21. Some dancers are over 40 Formal Formal negation Informal negation

22. ~(  x, P(x)  Q(x))  x such that ~(P(x)  Q(x))  x such that P(x)  ~Q(x)

23. More on Universal Conditional  x  U if P(x) then Q(x) Contrapositive Converse Conditional

24. Necessary and Sufficient Conditions, Only If  x, r(x) is a sufficient condition for s(x).  x, r(x) is a necessary condition for s(x).  x, r(x) only if s(x).