White MT_sln - Prof Bjorn Poonen MATH 55 MIDTERM SOLUTIONS(white(1(5 pts each For each of(a(g below If the proposition is true write TRUE If the

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Prof. Bjorn Poonen October 16, 2001 MATH 55 MIDTERM SOLUTIONS (white) (1) (5 pts. each) For each of (a)-(g) below: If the proposition is true, write TRUE. If the proposition is false, write FALSE. (Please do not use the abbreviations T and F, since in handwriting they are sometimes indistiguishable.) No explanations are required in this problem. (a) The value of ( - 141) mod 8 is - 5 . FALSE. The value of ( - 141) mod 8 is an integer between 0 and 7. (b) The function 3 n 2 log n + 5 n (log n ) 4 is O ( n 3 ) . TRUE. If c > 0, then log n is O ( n c ), so 3 n 2 log n is O ( n 2+ c ) and 5 n (log n ) 4 is O ( n 1+4 c ). Thus 3 n 2 log n + 5 n (log n ) 4 is O (max { n 2+ c , n 1+4 c } ) for any c > 0. In particular, this holds for c = 1 / 2, in which case we find that 3 n 2 log n + 5 n (log n ) 4 is O ( n 3 ). (c) The ceiling function f ( x ) = d x e , considered as a function from R to R , has an inverse function. FALSE. The ceiling function is not injective since b 1 c = b 1 / 2 c , so it is definitely not bijective. Hence it has no inverse function. (d) The set { 1 , 2 , 3 } × Z is countable. TRUE. One can list its elements in a sequence as follows: (1 , 0) , (2 , 0) , (3 , 0) , (1 , 1) , (2 , 1) , (3 , 1) , (1 , - 1) , (2 , - 1) , (3 , - 1) , (1 , 2) , (2 , 2) , (3 , 2) , (1 , - 2) , . . . (e) The proposition x y ( x y ) is true, when the universe of discourse is the set of natural numbers. TRUE. In words, this says “For all natural numbers
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This note was uploaded on 03/08/2010 for the course MATH 55 taught by Professor Strain during the Spring '08 term at University of California, Berkeley.

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White MT_sln - Prof Bjorn Poonen MATH 55 MIDTERM SOLUTIONS(white(1(5 pts each For each of(a(g below If the proposition is true write TRUE If the

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