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cs70_fa07_mt1_sol

# cs70_fa07_mt1_sol - CS70 Discrete Mathematics for Computer...

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CS70 Discrete Mathematics for Computer Science, Fall 2007 Midterm 1 Solutions Note: These solutions are not necessarily model answers. Rather, they are designed to be tutorial in nature, and sometimes contain more explanation (occasionally much more) than an ideal solution. Also, bear in mind that there may be more than one correct solution. The maximum total number of points available is 60. 1. Quick Questions (a) The truth tables are as follows: 6pts P Q P Q Q P P Q true true true true true true false false true false false true true false false false false true true true Almost all people got this question right. The most common mistake was in the table for P Q . (b) (i) and (iii) are valid strategies. (ii) and (iv) are invalid strategies. Although they were not required, 8pts here are explanations for these answers: The original statement is ( x P ( x )) ( y Q ( y )) . (i): The contrapositive is ( y ¬ Q ( y )) ( x ¬ P ( x )) . The contrapositive is logically equivalent to the original statement. Thus, we can approach this problem by assuming the left-hand side is true and showing that this implies the right-hand side. (ii): This would prove the statement ( y Q ( y )) ( x P ( x )) . This is the converse of the origi- nal statement which is not logically equivalent to the original statement. (iii): This is a proof by contradiction . It approaches the proof by assuming that the negation of the statement is true. The negation of the statement is ¬ [( x P ( x )) ( y Q ( y ))] ≡ ¬ [ ¬ ( x P ( x )) ( y Q ( y ))] ≡ ¬ [( x ¬ P ( x )) ( y Q ( y ))] ≡ ¬ ( x ¬ P ( x )) ∧ ¬ ( y Q ( y )) ( x P ( x )) ( y ¬ Q ( y )) (iv): This would prove the statement ¬ [( x P ( x )) ( y ¬ Q ( y ))] ≡ ¬ ( x P ( x )) ∨ ¬ ( y ¬ Q ( y )) ≡ ¬ ( x P ( x )) ( y Q ( y )) ( x P ( x )) ( y Q ( y )) which is not logically equivalent to the original statement. A number of people had some trouble with one or more parts of this problem. Some people reduced incorrectly. It was not necessary to show your work for this part.

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(c) The stable marriage produced by the algorithm is (1,B), (2,C), (3,A), (4,D). 4pts Almost all people got this question right. (d) We use the extended-gcd algorithm:
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cs70_fa07_mt1_sol - CS70 Discrete Mathematics for Computer...

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