03_Logic_1.6-1.8 IG a.pptx

Dx x is odd the domain for x is the set of all

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D(x): x is odd The domain for x is the set of all positive integers Is the following proposition true or false? x ((x < 0) → P(x)) A) True B) False

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Evaluating Quantified Statements Predicates A(x), B(x), C(x) Domain of x = {1, 2, 3, 4} x A(x) B(x) C(x) 1 T T T 2 T F T 3 T F F 4 T F F C(2) A(3) → B(3) B(3) v A(2) C(3) → A(3) x B(x) x ( A(x) ᴧ B(x) ᴧ C(x) ) x ( A(x) v C(x) ) x ( B(x) v (¬A(x) ᴧ C(x) )
Evaluating Quantified Statements Predicates A(x), B(x), C(x) Domain of x = {1, 2, 3, 4} x A(x) B(x) C(x) 1 T T T 2 F F T 3 T F F 4 T F F A) x ( A(x) → C(x) ) True x ( A(x) → C(x) ) True B) x ( A(x) → C(x) ) True x ( A(x) → C(x) ) False C) x ( A(x) → C(x) ) False x ( A(x) → C(x) ) True D) x ( A(x) → C(x) ) False x ( A(x) → C(x) ) False Which option is correct?

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Translating English to Logic Domain: students at a school S(x): x is a senior L(x): x was late to school Someone was not late to school Sam is a senior who was late to school Everyone was on time for school
Translating English to Logic Domain: students at a school S(x): x is a senior L(x): x was late to school There is a senior who was late to school Every student who is a senior was late to school.

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Logical Equivalence of Quantified Statements x ( S(x) ᴧ L(x) ) There is a senior who was late to school. Two quantified statements (whether they are expressed in English or the language of logic) have the same logical meaning if they have the same truth value regardless of value of the predicates for the elements in the domain.
Translating English to Logic Domain: students at a school S(x): x is a senior L(x): x was late to school Every student who is a senior was late to school. x S(x) L(x) Joe T F Belinda T F Lucy F F Brad F T x S(x) L(x) Nancy T T Ingrid T T Jose F F Ralph F T School 1 School 2

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Translating English to Logic Domain: students at a school S(x): x is a senior L(x): x was late to school There is a senior who was late to school. x S(x) L(x) Joe T F Belinda T F Lucy F F Brad F T x S(x) L(x) Nancy T T Ingrid T T Jose F F Ralph F T School 1 School 2
Translating English to Logic Domain: students at a school S(x): x is a senior F(x): x lives far from school L(x): x was late to school Select the logical expression that is equivalent to: Everyone who was late is a senior or lives far from school. A) x ( L(x) ᴧ (F(x) v S(x)) ) B) x ( L(x) → (F(x) v S(x)) ) C) x ( (L(x) ᴧ F(x)) v S(x) ) D) x ( (L(x) → F(x)) v S(x) )

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Logical Equivalence of Quantified Statements x C(x) ᴧ x D(x) x (C(x) ᴧ D(x)) ≡ (C(Bill) ᴧ D(Bill)) v (C(Frank) ᴧ D(Frank)) v (C(Margie) ᴧ D(Margie)) Someone has a cat and someone has a dog Someone has a cat and a dog x C(x) D(x) Bill T F Frank F T Margie T F x (C(x) ᴧ D(x))
De Morgan’s Law for Quantified Statements ¬ x L(x) x ¬L(x) There is no student who was late.

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• Spring '09
• Meenakshisundaram

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