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cs70_sp08_mt1_sol

# cs70_sp08_mt1_sol - CS 70 Spring 2008 Discrete Mathematics...

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CS 70 Discrete Mathematics for CS Spring 2008 David Wagner MT 1 Solns Midterm 1 Sample Solutions CS 70, Spring 2008, MT 1 Solns 1

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Problem 1. [True or false] (20 points) (a) TRUE or FALSE: Let the logical proposition R ( x ) be given by x 2 = 4 = x 1. Then R ( 3 ) is true. (False implies anything.) (b) TRUE or FALSE: The proposition P = ( P Q ) is logically equivalent to P = Q . (c) TRUE or FALSE : The proposition P = ( P Q ) is logically equivalent to ( P Q ) = P . (Consider P = True, Q = False.) (d) TRUE or FALSE: The proposition ( P Q ) ( ¬ P ∨¬ Q ) is a tautology, i.e., is logically equivalent to True. (e) TRUE or FALSE : n N . ( P ( n ) Q ( n )) is logically equivalent to ( n N . P ( n )) ( n N . Q ( n )) . (Consider the propositions P ( n ) = “n is odd” and Q ( n ) = “n is even”.) (f) TRUE or FALSE: n N . ( P ( n ) Q ( n )) is logically equivalent to ( n N . P ( n )) ( n N . Q ( n )) . (g) TRUE or FALSE: n N . (( k N . n = 2 k ) ( k N . n = 2 k + 1 )) . (Every natural number is either odd or even.) (h) TRUE or FALSE : n N . (( k N . n = 2 k ) ( k N . n = 2 k + 1 )) . (For any n N , take k = 100 n + 100 ; then n n = 2 k and n n = 2 k + 1 .) (i) TRUE or FALSE: n N . (( k N . n = k 2 ) = ( N . n = i = 1 ( 2 i 1 ))) . (For any n
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cs70_sp08_mt1_sol - CS 70 Spring 2008 Discrete Mathematics...

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