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Unformatted text preview: SOLUTIONS  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 ). Solution. [Similar to homework, 10 marks.] We prove the law using the following truth table. A B ¬ A ¬ B A ⇒ B ¬ B ⇒ ¬ A ( A ⇒ B ) ⇔ ¬ B ⇒ ¬ A T T F F T T T T F F T F F T F T T F T T T F F T T T T T A2. Let A , B , C be sets. Prove that A ( B ∩ C ) = ( A B ) ∪ ( A C ) . Solution. [Similar to homework, 10 marks.] { x ∈ A ( B ∩ C ) } ⇔ { x ∈ A ∧ ¬ ( x ∈ B ∩ C ) } ⇔ { ( x ∈ A ) ∧ [ ¬ ( x ∈ B ) ∨ ¬ ( x ∈ C )] } ⇔ { [( x ∈ A ) ∧ ¬ ( x ∈ B )] ∨ [( x ∈ A ) ∧ ¬ ( x ∈ C )] } ⇔ { ( x ∈ A B ) ∨ ( x ∈ A C ) } ⇔ { x ∈ ( A B ) ∪ ( A C ) } ....
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This note was uploaded on 01/29/2012 for the course MATH 11006 taught by Professor Ivormcgillivray during the Fall '08 term at University of Bristol.
 Fall '08
 IvorMcGillivray

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