Lec05 - Discrete Mathematics Rules of Inference and Proofs...

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Mehrdad Nojoumian Mehrdad Nojoumian Department of Computer Science Southern Illinois University CS 215: Discrete Mathematics Discrete Mathematics: Rules of Inference and Proofs
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Sec$on’Summary’ ± Inference’Rules’for’Propositional’Logic’ ± Using’Rules’of’Inference’to’Build’Arguments’ ± Rules’of’Inference’for’QuantiFed’Statements’ ± Building’Arguments’for’QuantiFed’Statements’ ± Mathematical’Proofs’ ± ±orms’of’Theorems’ ± Direct’Proofs’ ± Indirect’Proofs’ ± Proof’of’the’Contrapositive’ ± Proof’by’Contradiction’
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Argument:’Socrates’Example’ ± We’can’express’the’ premises ’(above’the’line)’and’ the’ conclusion ’(below’the’line)’in’predicate’logic’ as’an’argument:’ ± We’will’see’shortly’that’this’is’a’valid’argument.’
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Valid’Arguments’ ± We’will’show’how’to’construct’valid’arguments’in’two’ stages;’frst’For’ propositional logic ’and’then’For’ predicate logic .’The’ rules of inference are’the’essential’ building’block’in’the’construction’oF’valid’arguments.’ 1. Propositional’Logic’ ’ InFerence’Rules’ 2. Predicate’Logic’ ’ InFerence’rules’For’propositional’logic’plus’additional’ inFerence’rules’to’handle’variables’and’quantifers.’
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Arguments’in’Proposi$onal’Logic’ ± An’ argument in’propositional’logic’is’a’sequence’of’propositions.’ All’but’the’Fnal’proposition’are’called’ premises .’The’last’statement’ is’the’ conclusion . ± The’argument’is’ valid ’if’the’premises’ imply ’the’conclusion.’An’ argument form ’is’an’argument’that’is’valid’no’matter’what’ propositions’are’substituted’into’its’propositional’variables. ± If’the’premises’are’ p 1 , p 2 , …, p n and’the’conclusion’is’ q ’then’ ’( p 1 ∧ p 2 ∧ … ∧ p n ’)’ → q ’is’a’tautology. ± Inference rules ’are’“simple”’argument’forms’that’will’be’used’to’ construct’more’“complex”’argument’forms.
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Rules’of’Inference’for’Proposi$onal’ Logic:’Modus’Ponens’ Example: Let’ p ’be’“It’is’snowing.”’ Let’ q ’be’“I’will’study’discrete’math.”’ If ’it’is’snowing,’ then ’I’will’study’discrete’math.”’ “It’ is ’snowing.”’ Therefore ,’I’will’study’discrete’math.” Corresponding Tautology: (p’ ∧ (p →q)) → q
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Modus’ Tollens’ Example :’ Let’ p ’be’“it’is’snowing.”’ Let’ q ’be’“I’will’study’discrete’math.”’ If ’it’is’snowing,’ then ’I’will’study’discrete’math.”’ “I’will not ’study’discrete’math.”’ Therefore ,’it’is’ not ’snowing.” Corresponding Tautology: ( ¬ p’ ∧ (p →q)) → ¬q
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Lec05 - Discrete Mathematics Rules of Inference and Proofs...

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