Introduction to Predicates and Quantified Statements IINegation of a Universal StatementSlide 3Negations of Universal StatementsNegations of Existential StatementsExampleThe Word “Any”Negation of a Universal Conditional StatementSlide 9Negations and DeMorgan’s LawsSlide 11Evidence Supporting Universal StatementsSupporting Universal StatementsSlide 14Algebra PuzzlerIntroduction to Predicates and Quantified Statements IILecture 10Section 2.2 Fri, Feb 2, 2007Negation 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, x2 > 10” is false, then “x  R, x2  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, x2 < 0” is false, then “ x  R, x2  0” is true.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.”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 DeMorgan’s LawsLet the domain be D = {x1, x2, …, xn}.The statement x  D, P(x) is equivalent toP(x1)  P(x2)  …  P(xn).It’s negation isP(x1)  P(x2)  …  P(xn),which is equivalent tox  D, P(x).Negations and DeMorgan’s LawsThe statement x  D , P(x) is equivalent toP(x1)  P(x2)  …  P(xn).It’s negation isP(x1)  P(x2)  …  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)orC(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