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Somebodyislovedbyeverybody

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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 (...
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