Introduction to Predicates and Quantified Statements II

1. Introduction to Predicates and Quantified Statements II Lecture 10 Section 2.2 Fri, Feb 2, 2007

2. Negation of a Universal Statement • What would it take to make the statement “Everybody likes me” false?

3. Negation of a Universal Statement • What would it take to make the statement “Somebody likes me” false?

4. Negations of Universal Statements • The negation of the statement x  S, P(x) is the statement x  S, P(x). • If “x  R, x2 > 10” is false, then “x  R, x2  10” is true.

5. Negations of Existential Statements • The negation of the statement x  S, P(x) is the statement x  S, P(x). • If “x  R, x2 < 0” is false, then “x  R, x2  0” is true.

6. Example • Are these statements equivalent? • “Any investment plan is not right for all investors.” • “There is no investment plan that is right for all investors.”

7. The Word “Any” • We should avoid using the word “any” when writing quantified statements. • The meaning of “any” is ambiguous. • “You can’t put any person in that position and expect him to perform well.”

8. Negation of a Universal Conditional Statement • How would you show that the statement “You can’t get a good job without a good edikashun” is false?

9. Negation of a Universal Conditional Statement • The negation of x  S, P(x)  Q(x) is the statement x  S, (P(x)  Q(x)) which is equivalent to the statement x  S, P(x)  Q(x).

10. Negations and DeMorgan’s Laws • Let the domain be D = {x1, x2, …, xn}. • The statement x  D, P(x) is equivalent to P(x1)  P(x2)  …  P(xn). • It’s negation is P(x1)  P(x2)  …  P(xn), which is equivalent to x  D, P(x).

11. Negations and DeMorgan’s Laws • The statement x  D, P(x) is equivalent to P(x1)  P(x2)  …  P(xn). • It’s negation is P(x1)  P(x2)  …  P(xn), which is equivalent to x  D, P(x).

12. Evidence Supporting Universal Statements • Consider the statement “All crows are black.” • Let C(x) be the predicate “x is a crow.” • Let B(x) be the predicate “x is black.” • The statement can be written formally as x, C(x)  B(x) or C(x)  B(x).

13. Supporting Universal Statements • Question: What would constitute statistical evidence in support of this statement?

14. Supporting Universal Statements • The statement is logically equivalent to x, ~B(x)  ~C(x) or ~B(x)  ~C(x). • Question: What would constitute statistical evidence in support of this statement?

15. Algebra Puzzler • Find the error(s) in the following “solution.”