Analysis Mid Sessional Prac Paper

Analysis Mid Sessional Prac Paper - A is bounded above”(b...

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ANALYSIS - MIDSESSIONAL EXAM JANUARY 2008 This paper has two sections, A an B. Please use a separate answer book for each section. Section A has four short questions, ALL of which you should answer. Each of these questions is worth 10 marks. Section B has two longer questions, of which you should answer ONE . Each of these questions is worth 60 marks. The total mark for the paper is 100. Calculators MAY NOT be used in this examination. NOTE: The notation N + stands for the set of all positive integers: N + = { 1 , 2 , 3 ,... } . 1. Short Questions A1. Using a truth table, prove the contrapositive law ( A B ) ( ¬ B ⇒ ¬ A ). A2. Let A , B , C be sets. Prove that A - ( B C ) = ( A - B ) ( A - C ) . A3. Prove that 5 is irrational. A4. Let A R be a set. State a precise definition of each of the following: (a) “the set
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Unformatted text preview: A is bounded above”; (b) “the set A is not bounded above”; (c) a = sup A . 2. Longer Questions B1. Let f : X → Y be a function. (a) State the definitions of “ f is an injection” and “ f is a surjection”. (b) Prove that if f is a bijection then f is invertible. (c) Prove that the function f : R- {-2 } → R- {-1 } , f ( x ) = 3-x x + 2 is a bijection and find its inverse. B2. (a) State and prove the Archimedian Principle. (You may use without proof the theorem on existence of the supremum for a set of real numbers which is bounded above.) (b) Let A = ± 3 n-6 n | n ∈ N + ² . Find the supremum of the set A ? Justify your answer using the Archimedian Principle. 1...
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