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l19soln

# l19soln - CS 245 Winter 2009 Lecture 19 Suggested Exercises...

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CS 245 Winter 2009 Lecture 19 Suggested Exercises Solution Set Exercise 1 1. Every person enjoys a good time. There is a student. Therefore, some student enjoys a good time. We formalize this is, adding the hidden premise that all students are people, as: x person ( x ) good time ( x ) x student ( x ) Hidden Premise: x student ( x ) person ( x ) Therefore, x student ( x ) good time ( x ) where person ( x ) – means that x is a person, good time ( x ) – means that x enjoys a good time, and student ( x ) – means that x is a student. A proof of the argument is as follows: 1 x person ( x ) good time ( x ) premise 2 x student ( x ) premise 3 x student ( x ) person ( x ) Hidden premise 4 x u student ( x u ) assumption 5 student ( x u ) person ( x u ) E 3 6 person ( x u ) E 4 , 5 7 person ( x u ) good time ( x u ) E 1 8 good time ( x u ) E 6 , 7 9 student ( x u ) good time ( x u ) I 4 , 8 10 x student ( x ) good time ( x ) I 9 11 x student ( x ) good time ( x ) E 2 , 4 10 1

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2. There is someone in the first row who is in front of everyone else in the first row. Therefore, there is at most one person in the first row who is in front of everyone else in the first row. We formalize this is as: x R ( x ) ∧ ∀ y R ( y ) ∧ ¬ ( x = y ) F ( x,y ) Therefore, x • ∀ y R ( x ) R ( y ) ( z 1 R ( z 1 ) ∧ ¬ ( y = z 1 ) F ( y,z 1 )) ( z 2 R ( z 2 ) ∧ ¬ ( z 2 = x ) F ( x,z 2 )) ( y = x ) and add the hidden premise that no one can be both in front of someone and also be behind that same person: Hidden Premise: x • ∀ y • ¬ ( F ( x,y ) F ( y,x )) where R ( x ) – means that x is in the front row and F ( x,y ) – means that y is behind x . That is, x is in front of y .
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l19soln - CS 245 Winter 2009 Lecture 19 Suggested Exercises...

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