05h - Need practice Laws are not clear Discrete Mathematics...

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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 square4 double negation square4 DeMorgan’s laws square4 commutative, associative, and distributive laws square4 idempotent, identity, and domination laws square4 the law of contradiction and the law of excluded middle square4 absorption laws Discrete Mathematics – Rules of Inference 4-3 Expressing Connectives Some connectives can be expressed through others square4 p q ¬ (p q) square4 p q (p q) (q p) square4 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, 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 .
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