05a-Rules-of-Inference

05a-Rules-of-Inference - Rules of Introduction Inference...

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Rules of Inference Discrete Mathematics
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Discrete Mathematics – Rules of Inference 5-2 Previous Lecture Logically equivalent statements Statements Φ and Ψ are equivalent iff Φ↔Ψ is a tautology Main logic equivalences double negation DeMorgan’s laws commutative, associative, and distributive laws idempotent, identity, and domination laws the law of contradiction and the law of excluded middle absorption laws
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Discrete Mathematics – Rules of Inference 5-3 First Law of Substitution Suppose that the compound statement Φ is a tautology. If p is a primitive statement that appears in Φ and we replace each occurrence of p by the same statement q, then the resulting compound statement Ψ is also a tautology. Let Φ = (p q) (q p), and we substitute p by p (s r) Therefore ((p (s r)) q) (q (p (s r)) is a tautology
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Discrete Mathematics – Rules of Inference 5-4 Second Law of Substitution Let Φ be a compound statement, p an arbitrary (not necessarily primitive!) statement that appears in Φ , and let q be a statement such that p q. If we replace one or more occurrences of p by q, then for the resulting compound statement Ψ we have Φ Ψ . Let Φ = (p q) (q p), and we substitute the first occurrence of p by p (p q). Recall that p p (p q) by Absorption Law. Therefore (p q) (q p) ((p (p q)) q) (q p).
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Discrete Mathematics – Rules of Inference 5-5 Logic Inference One of the main goals of logic is to distinguish valid and invalid arguments What can we say about the following arguments: ``If you have a current password, then you can log onto the network.
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This note was uploaded on 12/12/2010 for the course MACM 201 taught by Professor Marnimishna during the Fall '09 term at Simon Fraser.

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05a-Rules-of-Inference - Rules of Introduction Inference...

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