klarfeld-final-exam - Jennifer Klarfeld Final Exam 1 For...

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Jennifer’Klarfeld’ –’ Final’ Exam’ ’ 1 1. For’propositions’ ,’ ± ,’and’ ² ,’prove’that’ ~(࠵ ∧ ±) ’ and’ ± → ~࠵ ’ are’equivalent. ’ ± ࠵ ∧ ± ~(࠵ ∧ ±) ~࠵ ± → ~࠵ T T T F’ F F’ F T F T’ T T’ T F F T’ F T’ F F F T’ T T’ As’we’can’see’from’the’above’truth’table,’ ~(࠵ ∧ ±) ’ and’ ± → ~࠵ ’ have’the’same’truth’ values’when’they’have’the’same’correspondi ng’truth’values’for’ ’ and’ ± .’Thus,’the’ two’statements’are’equivalent.
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Jennifer’Klarfeld’ –’ Final’ Exam’ ’ 2 2. Let’ ’ be’the’universe,’and’let’ ± ,’ ² ,’and’ ³ ’ be’subsets’of’ .’Prove’that: ’ (a) ± ∪ ² ∩ ³ = (± ∪ ²) ∩ (± ∪ ³) ´ ∈ ± ∪ ² ∩ ³ ’ iff’ ´ ∈ ± ’ or’ ´ ∈ ² ∩ ³ iff’ ´ ∈ ± ’ or’ (´ ∈ ² ’ and’ ´ ∈ ³) iff’ (´ ∈ ± ’ or’ ´ ∈ ²) ’ and’ (´ ∈ ± ’ or’ ´ ∈ ³) iff’ ´ ∈ ± ∪ ² ’ and’ ´ ∈ ± ∪ ³ ’ iff’ ´ ∈ ± ∪ ² ± ∪ ³ . Thus,’ ± ∪ ² ∩ ³ = (± ∪ ²) ∩ (± ∪ ³) .’ (b) ± − ² = ± ∩ ² ³ ´ ∈ ± − ² ’ iff’ ´ ∈ ± ’ and’ ´ ∉ ² iff’ ´ ∈ ± ’ and’ ´ ∈ ² ³ iff’ ´ ∈ ± ∩ ² ³ . Thus,’ ± − ² = ± ∩ ² ³ .’
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Jennifer’Klarfeld’ –’ Final’ Exam’ ’ 3 3. Use’the’Principle’of’Mathematical’Induction’(PMI)’to’prove’that’for’all’ ࠵ ∈ ℕ ’ with’ ’ ࠵ ≥ ± ,’ ± ² − ³ ² + ³ ²=± ´ µ࠵ + ± ࠵ − ³ ±࠵ ࠵ + ³ (i) Let’ ࠵ ´ ± .’ Then’ ± ²?³ ²@³ ²=± ´ ± ±?³ ±@³ ´ ± µ ´ ³± ´ ³ · µ ´ µ ± ±?³ ± ± ±@³ . ’ ’ Thus,’ ± ²?³ ²@³ ²=± ´ µ࠵@± ࠵?³ ±࠵ ࠵@³ ’ for’ ࠵ ´ ±. (ii) Now’assu me’that’ ± ²?³ ²@³ ²=± ´ µ࠵@± ࠵?³ ±࠵ ࠵@³ ’ for’all’ ࠵ ∈ ℕ,࠵ ≥ ±. Then’ ± ²?³ ²@³ ࠵@³ ²=± ´ ± ²?³ ²@³ ²=± + ± ࠵@³ ࠵@³
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