quiz2bANS - ¬∃ x ( H ( x ) ∧ A ( x )) ≡ ∀ x ( ¬ H...

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Math 240, Quiz 1 Name: Circle One: T 12:05 T 2:25 R 12:05 R 2:25 Instructions: Answer all questions fully, showing work where necessary. 1) Show that ( p q ) ( p r ) and( p ( q r ) are logically equivalent. The truth table is this: p q r p q p r ( p q ) ( p r ) q r p ( q r ) ( p q ) ( p r ) ( p ( q r ) T T T T T T T T T T T F T F T T T T T F T F T T T T T T F F F F F F F T F T T T T T T T T F T F T T T T T T F F T T T T T T T F F F T T T F T T 2)Express each of these statemtns using quanti±ers. Then form the negation of the statment so that no negation is to the left of a quanti±er. Next, express the negation in simple English (Do not simply use the words “It is not the case that”) a) There is a horse that can add. Let H ( x ) mean “ x is a horse”, and A ( x ) mean “ x can add”. Then we write this as x ( H ( x ) A ( x ). The negation is
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Unformatted text preview: ¬∃ x ( H ( x ) ∧ A ( x )) ≡ ∀ x ( ¬ H ( x ) ∧ A ( x )) ≡ ∀ x ( ¬ H ( x ) ∨ ¬ A ( x ). Of course, there are other things this is equivalent to. A correct translation might be “No horse can add”. b) Every koala can climb. Let K ( x ) mean “ x can add”, C ( x ) mean “ x can climb”. Then we can interpret this statement as ∀ x ( K ( x ) → C ( x )) The negation is ¬∀ x ( K ( x ) → C ( x )) ≡ ∃ x ¬ ( K ( x ) → C ( x )) ≡ ∃ x ( K ( x ) ∧ C ( x )). A proper translation might be “There exists a koala that cannot climb”. 1...
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This note was uploaded on 08/11/2008 for the course MATH 240 taught by Professor Miller during the Fall '08 term at University of Wisconsin.

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