practice_for_final - PRACTICE PROBLEMS FOR THE FINAL Math...

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PRACTICE PROBLEMS FOR THE FINAL Math 444, Fall 2014 Anush Tserunyan miscellaneous topics Problem 1. (20 points) Carefully read each of the following statements and circle T if the statement is TRUE and F if it is FALSE (you don’t have to justify your answer): (i) Every surjective function has an inverse. T F (ii) There is a bijection between Q and N . T F (iii) sup { 2 } { 1 - 1 n n N } = 1. T F (iv) Every bounded set contains its supremum. T F (v) If a sequence is bounded, it must converge. T F (vi) If a sequence ( x n ) is decreasing and bounded, then lim n x n = inf n N x n . T F (vii) If every bounded subsequence converges then the sequence itself converges. T F sets and functions Problem 2. (a) Let A,B be sets and a A,b B . Prove that if there is a bijection f A B , then there is a (perhaps different) bijection g A B such that g ( a ) = b . Hint : Think about A being a set of students, B being a set of chairs, a a particular student and b a particular chair. We are given that there is a way to sit students in the chairs so that every student gets exactly one seat and every chair is occupied by exactly one student. How can we modify the student–chair assignment so that the student a gets the chair b ? (b) Let n,m N . Prove that if { 1 , 2 ,...,n,n + 1 } { 1 , 2 ,...,m } , then { 1 , 2 ,...,n } { 1 , 2 ,...,m - 1 } , and in particular, m 2.
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