Nested Quantifiers CSAPMA 202 Spring 2005 Rosen section

Nested Quantifiers CS/APMA 202, Spring 2005 Rosen, section 1. 4 Aaron Bloomfield 1

Multiple quantifiers You can have multiple quantifiers on a statement x y P(x, y) n n “For all x, there exists a y such that P(x, y)” Example: x y (x+y == 0) x y P(x, y) n n There exists an x such that for all y P(x, y) is true” x y (x*y == 0) 2

Order of quantifiers x y and x y are not equivalent! x y P(x, y) n P(x, y) = (x+y == 0) is false x y P(x, y) n P(x, y) = (x+y == 0) is true 3

Negating multiple quantifiers Recall negation rules for single quantifiers: n n n ¬ x P(x) = x ¬P(x) Essentially, you change the quantifier(s), and negate what it’s quantifying Examples: n ¬( x y P(x, y)) = x ¬ y P(x, y) = x y ¬P(x, y) n ¬( x y z P(x, y, z)) = x¬ y z P(x, y, z) = x ¬y z P(x, y, z) = x y z ¬P(x, y, z) 4

Negating multiple quantifiers 2 Consider ¬( x y P(x, y)) = x y ¬P(x, y) n n The left side is saying “for all x, there exists a y such that P is true” To disprove it (negate it), you need to show that “there exists an x such that for all y, P is false” Consider ¬( x y P(x, y)) = x y ¬P(x, y) n n The left side is saying “there exists an x such that for all y, P is true” To disprove it (negate it), you need to show that “for all x, there exists a y such that P is false” 5

Translating between English and quantifiers Rosen, section 1. 4, question 20 The product of two negative integers is positive n n x y ((x<0) (y<0) → (xy > 0)) Why conditional instead of and? The average of two positive integers is positive n x y ((x>0) (y>0) → ((x+y)/2 > 0)) The difference of two negative integers is not necessarily negative n n x y ((x<0) (y<0) (x-y≥ 0)) Why and instead of conditional? The absolute value of the sum of two integers does not exceed the sum of the absolute values of these integers n x y (|x+y| ≤ |x| + |y|) 6

Translating between English and quantifiers Rosen, section 1. 4, question 24 x y (x+y = y) n There exists an additive identity for all real numbers x y (((x≥ 0) (y<0)) → (x-y > 0)) n A non-negative number minus a negative number is greater than zero x y (((x≤ 0) (y≤ 0)) (x-y > 0)) n The difference between two non-positive numbers is not necessarily non-positive (i. e. can be positive) x y (((x≠ 0) (y≠ 0)) ↔ (xy ≠ 0)) n The product of two non-zero numbers is non-zero if and only if both factors are non-zero 7

Rosen, section 1. 4 question 30 Rewrite these statements so that the negations only appear within the predicates a) y x P(x, y) b) x y P(x, y) c) y (Q(y) x R(x, y)) y ( Q(y) ( x R(x, y))) y ( Q(y) x R(x, y)) 9

Rosen, section 1. 4 question 31 a) Express the negations of each of these statements so that all negation symbols immediately precede predicates. x y z T(x, y, z) b) x y P(x, y) x y Q(x, y) ( x y z T(x, y, z)) x y z T(x, y, z) ( x y P(x, y) x y Q(x, y)) x y P(x, y) x y Q(x, y) x y P(x, y) x y Q(x, y) 10

Quick survey n a) b) c) d) I felt I understood the material in this slide set… Very well With some review, I’ll be good Not really Not at all 11

Quick survey n a) b) c) d) The pace of the lecture for this slide set was… Fast About right A little slow Too slow 12

Quick survey n a) b) c) d) How interesting was the material in this slide set? Be honest! Wow! That was SOOOOOO cool! Somewhat interesting Rather borting Zzzzzz 13