Lecture5 - Discrete Mathematics Rules of Inference 5-2...

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Introduction Rules of Inference Discrete Mathematics Andrei Bulatov 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 s double negation s DeMorgan’s laws s commutative, associative, and distributive laws s idempotent, identity, and domination laws s the law of contradiction and the law of excluded middle s absorption laws Discrete Mathematics – Rules of Inference 4-3 Expressing Connectives Some connectives can be expressed through others s p q ⇔ ¬ (p q) s p q (p q) (q p) s p q ⇔ ¬ p q Theorem Every compound statement is logically equivalent to a statement that uses only conjunction, disjunction, and negation Discrete Mathematics – Rules of Inference 4-4 Example Simplify the statement (p q) (p q) Discrete Mathematics – Rules of Inference 5-5 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 (not necessarily primitive), 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 Discrete Mathematics – Rules of Inference 5-6 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
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Lecture5 - Discrete Mathematics Rules of Inference 5-2...

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