Introduction to Predicates and Quantified Statements II Lecture

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

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

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

Negations of Universal Statements The negation of the statement x S, P(x) is the statement x S, P(x). ¢ If “ x R, x 2 > 10” is false, then “ x R, x 2 10” is true. ¢

Negations of Existential Statements The negation of the statement x S, P(x) is the statement x S, P(x). ¢ If “ x R, x 2 < 0” is false, then “ x R, x 2 0” is true. ¢

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

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. ” ¢

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?

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).

Negations and De. Morgan’s Laws Let the domain be D = {x 1, x 2, …, xn}. ¢ The statement x D, P(x) is equivalent to P(x 1) P(x 2) … P(xn). ¢ It’s negation is P(x 1) P(x 2) … P(xn), which is equivalent to x D, P(x). ¢

Negations and De. Morgan’s Laws The statement x D, P(x) is equivalent to P(x 1) P(x 2) … P(xn). ¢ It’s negation is P(x 1) P(x 2) … P(xn), which is equivalent to x D, P(x). ¢

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). ¢

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

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? ¢

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