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Lecture6

# Lecture6 - Discrete Mathematics Logic Inference 6-2...

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Introduction Logic Inference Discrete Mathematics Andrei Bulatov Discrete Mathematics - Logic Inference 6-2 Previous Lecture Valid and invalid arguments Arguments and tautologies Rules of inference Discrete Mathematics – Logic Inference 6-3 General Definition of Inference The general form of an argument in symbolic form is q p p p p ) ( 3 2 1 n K premises conclusion The argument is valid if whenever each of the premises is true the conclusion is also true The argument is valid if and only if the following compound statement is a tautology q p p p p q p p p p ) ) ) ((( 3 2 1 3 2 1 n n K K Discrete Mathematics – Logic Inference 6-4 Rules of Inference Verifying if a complicated statement is a tautology is nearly impossible, even for computer. Fortunately, general arguments can be replaced with small collection of simple ones, rules of inference . p q p q Modus ponens ``If you have a current password, then you can log onto the network. You have a current password. Therefore, you can log onto the network.’’ Discrete Mathematics – Logic Inference 6-5 Rule of Syllogism p q q r p r ``If you send me an e-mail, then I’ll finish writing the program. If I finish writing the program, then I’ll go to sleep early.’’ p - `you will send me an e-mail’ q - `I will finish writing the program’ r - `I will go to sleep early’ ``Therefore, if you send me an e-mail, then I’ll go to sleep early’’ The corresponding tautology ((p q) (q r)) (p r) Discrete Mathematics – Logic Inference 6-6 Modus Tollens p q ¬ q ∴ ¬ p The corresponding tautology ((p q) ∧ ¬ q) → ¬ p ``If today is Friday, then tomorrow I’ll go skiing’’. ``I will not go skiing tomorrow’’. p - `today is Friday’ q - `I will go skiing tomorrow’ ``Therefore, today is not Friday’’

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Discrete Mathematics – Logic Inference 6-7 Rule of Disjunctive Syllogism p q ¬ p q
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Lecture6 - Discrete Mathematics Logic Inference 6-2...

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