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CS 341 Homework 1 Answers 1) Prove each of the following: a) (( A B ) C ) ( ¬ A ¬ B C ). (( A B ) C ) ( ¬ A ¬ B C ) ( ¬ ( A B ) C ) ( ¬ A ¬ B C ) Definition of (( ¬ A ¬ B ) C ) ( ¬ A ¬ B C ) de Morgan’s Law ( ¬ A ¬ B C ) ( ¬ A ¬ B C ) Associativity of True Definition of b) ( A ¬ B ¬ C ) ( A ¬ ( B C )). ( A ¬ B ¬ C ) ( A ¬ ( B C )) ¬ ( A ¬ B ¬ C ) ( A ¬ ( B C )) Definition of ¬ A B C ( A ¬ ( B C )) de Morgan’s Law ¬ A B C A ¬ ( B C ) Associativity of ¬ A B C A ¬ B ¬ C de Morgan’s Law ¬ A A B ¬ B C ¬ C Commutativity of ( ¬ A A ) ( B ¬ B ) ( C ¬ C ) Associativity of True True True Definition of True Definition of 2) List the elements of each of the following sets: a) P ({apple, pear, banana}). , {apple}, {pear}, {banana}, {apple, pear}, {apple, banana}, {pear, banana}, {apple, pear,banana} b) P ({ a , b }) - P ({ a , c }). { b }, { a , b } c) P ( ). d) { a , b } × {1, 2, 3} × . e) { x : ( x 7 x 7}. 7 f) { x : 5 y ( y < 10 ( y + 2 = x ))} (where is the set of nonnegative integers). 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 g) { x : 5 y ( 5 z   (( x = y + z ) ( y < 5) ( z < 4)))}. 0, 1, 2, 3, 4, 5, 6, 7 3) Prove each of the following: a) A ( B C D ) = ( A B ) ( A D ) ( A C ). A ( B C D ) = ( A B ) ( A D ) ( A C ). Set union distributes over set intersection.

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4) Consider the English sentence, “If some bakery sells stale bread and some hotel sells flat soda, then the only thing everyone likes is tea.” This sentence has at least two meanings. Write two (logically different) first-order- logic sentences that correspond to meanings that could be assigned to this sentence. Use the following predicates: B ( x ) is True iff x is a bakery; S B ( x ) is True iff x sells stale bread; H ( x ) is True iff x is a hotel; S S ( x ) is True iff x sells flat soda; L ( x , y
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