Midterm 2 Review the Exam Will Cover Chapters 4 (Functions
Midterm 2 Review The exam will cover Chapters 4 (Functions, injections, surjections, bijections, cardinality of sets etc.) and 13 (sequences, convergence, bounded, monotone etc.). 1 As I stated in class, the first problem on the test will be to state the definition of a sequence (an)n=1 converging to a limit L 2 R. You should also be able to state the other definitions in these chapters, and the exam will also ask for some of these definitions. Here is a list of definitions you should be able to state: • injective function • surjective function • bijective function • cardinality of a finite set • countable set • composition of two functions • convergent sequence • bounded sequence • monotone sequence • supremum and infimum of a set/sequence You should also be able to produce examples of each of these objects, and also examples which have various combinations of these properties (e.g. a function which is injective but not surjective, or a sequence which is bounded but not monotone), as well as when there exist no examples (e.g. an unbounded convergent sequence, which cannot exist because every convergent sequence is bounded). For example, if X = f1; 2; 3g and Y = f4; 5; 6; 7g, then an example of a function f : X ! Y which is injective and not surjective would be to define f as follows: f(1) = 5, f(2) = 4, f(3) = 7. n For example, a sequence which is bounded but not monotone is an = (−1) . You should be able to prove that a given sequence converges, and good practice for this would be (−1)n 1 1 1 p1 showing that sequences like n , n2 , n3 , n , n all converge.
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