sol1 - CS70 Discrete Mathematics and Probability, Fall 2009...

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CS70 Discrete Mathematics and Probability, Fall 2009 Homework 1 Solutions Note: These solutions are not necessarily model answers. Rather, they are designed to be tutorial in nature, and sometimes contain a little more explanation 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 40. 1. [Classical Logic] [15 pts] (a) It is useful to rephrase the sentences to be in the form “If X , then Y .” Writing sentences as quantified propositions: (I) “No shark ever doubts that he is well fitted out.” This means: “If a fish is a shark, then it does not doubt that it is well fitted out.” x,B ( x ) ⇒ ¬ P ( x ) . [1 pt ] (II) “A fish, that cannot dance a minuet, is contemptible.” x, ¬ U ( x ) F ( x ) . [1 pt ] (III) “No fish is quite certain that it is well fitted out, unless it has three rows of teeth.” This statement means that “A fish is certain that it is well fitted out only if it has three rows of teeth,” or equivalently, “If a fish is certain that it is well fitted out, then it has three rows of teeth.” Thus we have x, ¬ P ( x ) O ( x ) . [1 pt ] (IV) “All fishes, except sharks, are kind to children.” This means: “If a fish is not a shark, then it is kind to children.” x, ¬ B ( x ) N ( x ) . [1 pt ] (V) “No heavy fish can dance a minuet.” This means: “If a fish is heavy, then it cannot dance a minuet.” x,K ( x ) ⇒ ¬ U ( x ) . [1 pt ] (VI) “A fish with three rows of teeth is not to be despised.” x,O ( x ) ⇒ ¬ F ( x ) . [1 pt ] Many people got the wrong order of implication in parts I, III, and V, or got the negations wrong. Note that the contrapositive of each implication above is also an acceptable answer.
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This note was uploaded on 11/05/2009 for the course CS 70 taught by Professor Papadimitrou during the Spring '08 term at University of California, Berkeley.

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sol1 - CS70 Discrete Mathematics and Probability, Fall 2009...

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