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with UoD = nonÂnegative integers. âˆ€ âˆƒ 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.)
â€¢ Examples:
âˆƒ x red(x) = T for UoD = all apples
âˆƒ x red(x) = F for UoD = golden delicious apples Discussion #14 Chapter 2, Section 1 14/20 Expressions with Quantifiers â€¢ Quantifiers associate right to left. â€“ Example with UoD = {Ann, Sue, Tim}
âˆ€xâˆƒ y loves(x, y)
= (âˆ€x(âˆƒ y(loves(x, y))))
= âˆ€x(loves(x, Ann) âˆ¨ loves(x, Sue) âˆ¨ loves(x, Tim))
= (loves(Ann, Ann) âˆ¨ loves(Ann, Sue) âˆ¨ loves(Ann, Tim)) âˆ§(loves(Sue, Ann) âˆ¨ loves(Sue, Sue) âˆ¨ loves(Sue, Tim)) âˆ§(loves(Tim, Ann) âˆ¨ loves(Tim, Sue) âˆ¨ loves(Tim, Tim)) â€¢ 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 (continuedâ€¦)
â€¢ What about âˆƒ yâˆ€x loves(x, y)?
â€“
â€“
â€“ There exists a y such that for every x, x loves y. Somebody is loved by everybody.
Not the same as everybody loves somebody. â€¢ What about âˆƒ xâˆ€y loves(x, y)? â€“ There exists an x such that for every y, x loves y.
â€“ Somebody loves everybody. â€¢ What about âˆ€yâˆƒ x loves(x, y)? â€“ For every y there exists an x such that x loves y.
â€“ Everybody is loved by somebody. â€¢ Order matters.
Discussion #14 Chapter 2, Section 1 16/20 Precedence
Quantifiers have the highest precedence:
Â¬ âˆƒ âˆ€ (unary operators)
âˆ§
âˆ¨
â‡’
â‡” âˆ€yâˆƒ x P(x, y) âˆ¨ Q(x) âˆ§Â¬âˆƒ x R(x, y, z) âˆ€y (âˆƒ x P(x, y)) âˆ¨ Q(x) âˆ§Â¬(âˆƒ x R(x, y, z)) (âˆ€y (â...
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 Winter '12
 MichaelGoodrich
 Sue, Jim, Tim, Ann, Zed

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