14-PredCalc

14-PredCalc

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: … with UoD = non­negative integers. ∀ ∃ x P(x) = T when P(x) = T for one or more substitutions from UoD. (Only need one true predicate instantiation to make the formula true.) • Examples: ∃ x red(x) = T for UoD = all apples ∃ x red(x) = F for UoD = golden delicious apples Discussion #14 Chapter 2, Section 1 14/20 Expressions with Quantifiers • Quantifiers associate right to left. – Example with UoD = {Ann, Sue, Tim} ∀x∃ y loves(x, y) = (∀x(∃ y(loves(x, y)))) = ∀x(loves(x, Ann) ∨ loves(x, Sue) ∨ loves(x, Tim)) = (loves(Ann, Ann) ∨ loves(Ann, Sue) ∨ loves(Ann, Tim)) ∧(loves(Sue, Ann) ∨ loves(Sue, Sue) ∨ loves(Sue, Tim)) ∧(loves(Tim, Ann) ∨ loves(Tim, Sue) ∨ loves(Tim, Tim)) • We say, for every x, there exists a y such that x loves y. (Everybody loves somebody.) Discussion #14 Chapter 2, Section 1 15/20 Expressions with Quantificates (continued…) • What about ∃ y∀x loves(x, y)? – – – There exists a y such that for every x, x loves y. Somebody is loved by everybody. Not the same as everybody loves somebody. • What about ∃ x∀y loves(x, y)? – There exists an x such that for every y, x loves y. – Somebody loves everybody. • What about ∀y∃ x loves(x, y)? – For every y there exists an x such that x loves y. – Everybody is loved by somebody. • Order matters. Discussion #14 Chapter 2, Section 1 16/20 Precedence Quantifiers have the highest precedence: ¬ ∃ ∀ (unary operators) ∧ ∨ ⇒ ⇔ ∀y∃ x P(x, y) ∨ Q(x) ∧¬∃ x R(x, y, z) ∀y (∃ x P(x, y)) ∨ Q(x) ∧¬(∃ x R(x, y, z)) (∀y (...
View Full Document

This note was uploaded on 03/02/2012 for the course C S 236 taught by Professor Michaelgoodrich during the Winter '12 term at BYU.

Ask a homework question - tutors are online