Quesons on order of quaners example 2 let domain be

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Unformatted text preview: w we have: Lewis Carroll Example —༉  The first two are premises and the third is the conclusion. “All lions are fierce.” 2.  “Some lions do not drink coffee.” 3.  “Some fierce creatures do not drink coffee.” 1.  —༉  Let P(x), Q(x), and R(x) be the propositional functions “x is a lion,” “x is fierce,” and “x drinks coffee,” respectively. 1.  ∀x (P(x)→ Q(x)) 2.  ∃x (P(x) ∧ ¬R(x)) 3.  ∃x (Q(x) ∧ ¬R(x)) Nested Quan$fiers —༉  Nested quantifiers are often necessary to express the meaning of sentences in English as well as important concepts in computer science and mathematics. Example: “Every real number has an inverse” is ∀x ∃y (x + y = 0) where the domains of x and y are the real numbers. —༉  We can also think of nested propositional functions: ∀x ∃y(x + y = 0) can be viewed as ∀x Q(x) where Q(x) is ∃y P(x, y) where P(x, y) is (x + y = 0) Thinking of Nested Quan$fica$on —༉  To evaluate ∀ x ∀ y P(x, y), loop through the values of x : —༉  At each step, loop through the values for y. —༉  If for some pair of x and y, P(x, y) is false, ∀x ∀y P(x, y) is false and both the outer and inner loop terminate. —༉  To evaluate ∀ x ∃ y P(x, y), loop through the values of x: —༉  —༉  At each step, loop through the values for y. If no y is found such that P(x, y) is true, the outer loop terminates as ∀x ∃y P(x, y) has been shown to be false. Order of Quan$fiers Examples: 1.  Let P(x, y) be the statement “x + y = y + x.” Assume that domain is the real numbers. Then ∀x ∀y P(x, y) and ∀y ∀x P(x, y) have the same truth value. 2.  Let Q(x, y) be...
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