15b_00f_fe

# 15b_00f_fe - Math 15B Final Exam 8 December 2000 Please put...

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Math 15B Final Exam 8 December 2000 Please put your name and ID number on your blue book. The exam is CLOSED BOOK, but calculators ARE allowed. You must show your work to receive credit. 1. (60 pts) In each case, give an example or explain why none exists. (a) A tree with exactly nine vertices and exactly nine edges. (b) A permutation f on { 1 , 2 , 3 , 4 } such that f 100 6 = f . Notes : Remember that f 6 = g for functions with the same domain means there is at least one x such that f ( x ) 6 = g ( x ). Also remember that f 100 ( x ) means f ( f ( ··· f ( x ) )), NOT ( f ( x )) 100 . (c) A simple graph with exactly ﬁve vertices that has a cycle containing six edges. (d) A sample space U with a probability function P and two diﬀerent elements s and t of U such that P ( s ) > 1 / 2 and also P ( t ) > 1 / 2. Note : P ( s ) means the same thing as P ( { s } ). (e) A sample space U with a probability function P and two diﬀerent subsets S and T of U such that P ( S ) > 1 / 2 and also P ( T ) > 1 / 2. (f) Two functions f ( n ) and g ( n ) such that “ f ( n )is O ( g ( n ))” is TRUE and, at the same time ,“ g ( n O ( f ( n ))” is FALSE.

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## This note was uploaded on 06/17/2009 for the course MATH 15B taught by Professor Bender during the Fall '01 term at UCSD.

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15b_00f_fe - Math 15B Final Exam 8 December 2000 Please put...

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