Introduction to Predicates and Quantified Statements II

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Presentation transcript:

Introduction to Predicates and Quantified Statements II Lecture 8 Section 2.2 Mon, Jan 24, 2005

Negations of Universal Statements The negation of 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.

Negations of Existential Statements The negation of 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.

Example: Negation of a Universal Statement p = “Everybody likes me.” Express p as x  {all people}, x likes me. p is the statement x  {all people}, x does not like me. p = “Somebody does not like me.”

Example: Negation of an Existential Statement p = “Somebody likes me.” Express p as x  {all people}, x likes me. p is the statement x  {all people}, x does not like me. p = “Everyone does not like me.” p = “Nobody likes me.”

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 x  S, P(x)  Q(x).

Example Find the error(s) in the following “solution.”

The Operators, , , and  Let D = {x1, x2, …, xn}. The statement x  D, P(x) is equivalent to P(x1)  P(x2)  …  P(xn). The statement x  D, P(x) is equivalent to P(x1)  P(x2)  …  P(xn).

The Operators, , , and  That is why x  D, P(x) is false if P(x) is false for a single x  D. And that is why x  D, P(x) is true if P(x) is true for a single x  D.

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). Question: What would constitute evidence in support of this statement?

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