Nested QuantifiersNiloufar Shafiei - PDF document

Nested QuantifiersNiloufar Shafiei y (x + y = 0)Q(x) is P(x,y) is (x + y = 0) Q(x) 2Nested quantifiers (example)Translate the following statement into English. x Domain: real numbers 3Nested quantifiers (example)Translate the following statement into English. x y (x = - y)Domain: real numbers Translate the following statement into English. x y ((x � 0) (y ) (xy 0))Domain: real numbersis negative then xy is negative. Assume P(x,y) is (xy = yx).y P(x,y)domain: real numbersxy = yx.For every pair of real numbers x, y, xy = yx. universal quantifierswithout other quantifiers Assume P(x,y) is (xy = 6).y P(x,y)domain: integersThere is a pair of integers x, y for which xy = 6. existentialwithout other quantifiers canbe changed without changing the meaning y P(x,y)domain: real numbersFor all real numbers x there is a real number y such that x + yTrue(y = 10 - x)x P(x,y)domain: real numbersThere is a real number y such that for all real numbers x, x + yFalsex P(x,y) are not logically equivalent. z P(x,y,z)domain: real numbersFor all real numbers x and y there is a real number z such thatTruey P(x,y,z)domain: real numbersFalsez P(x,y,z) and of nested existential and 12Quantification of two variable y P(x,y) P(x,y) is true for every pair x,y. y P(x,y) 13Quantification of two variable y P(x,y) There is an x for which P(x,y) is true for every y. y P(x,y)When true?There is a pair x, y for which P(x,y) is true.When false?P(x,y) is false for every pair x, y. and aevery pair of , if both integers are “The sum of two positive integers is always positive.”“For every pair of integers, if both integers are positive,then the sum of them is positive.”“For all integers x, y, if x and y are positive, then x+y is “The sum of two positive integers is always positive.”“For all integers x, y, if x and y are positive, then x+y is y� ((x 0) � (y 0) �(x + y 0))domain: integers x y� (x + y 0)domain: positive integers “Every real number except zero has a multiplicative inverse.”that xy = 1. and a every “Every real number except zero has a multiplicative inverse.”xy = 1.“For every real number except zero, there is a multiplicative“For every real number x, if x 0, then there is a real numbery such that xy = 1.” “Every real number except zero has a multiplicative inverse.”xy = 1.“For every real number x, if x 0, then there is a real numbery such that xy = 1.” 0) y (xy = 1))domain: real numbers 20Nested quantifiers (example)Translate the following statement into English. x (C(x) y (C(y) F(x,y)))“For every student x, x has a computer or there is a student y“Every student has a computer or has a friend that has acomputer.” 21Nested quantifiers (example)Translate the following statement into English. x z ((F(x,y) F(x,z) (y z)) “There is a student x such that for all students y and all“There is a student none of whose friends are also friends with “If a person is a student and is computer science major, thenthis person takes a course in mathematics. ” x ((S(x) C(x)) y T(x,y)) “Everyone has exactly one best friend. ” For all x, there is y who is the best friend of x and for everyperson z, if person z is not person y, then z is not the best x y (B(x,y) z ((z y) B(x,z)) “Everyone has exactly one best friend. ” For all x, there is y who is the best friend of x and for everyperson z, if person z is not person y, then z is not the best x z ( (B(x,y) B(x,z)) (y = z) ) “There is a person who has taken a flight on every airline in the x f (F(x,f) A(f,a)) “There is a person who has taken a flight on every airline in the x f R(x,f,a)Domain of x: all peopleDomain of f: all flightsDomain of a: all airlines 27Negating quantified expressions(review) x P(x) x P(x) P(x) 28Negating nested quantifiersRules for negating statements involving asingle quantifiers can be applied forquantifiers. 29Negating nested quantifiers(example)What is the negation of the following statement? x x P(x)P(x) = y (x = -y)P(x)y (x = -y))(x = -y))y (x “There is not a person who has taken a flight on every airline.” f (F(x,f) F(x,f): x has taken flight f.A(f,a): flight f is on airline a. f (F(x,f) A(f,a)) x ¬ f (F(x,f) A(f,a)) x f (F(x,f) A(f,a)) A(f,a)) x F(x,f) 31Recommended exercises1,3,10,13,23,25,27,33,39