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5 - LECTURE 5 QUANTIFICATION(2.9-2.10 We must give lengthy...

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LECTURE 5: QUANTIFICATION ( § 2.9-2.10) We must give lengthy deliberation to what has to be decided once and for all. Publilius Syrus ( 100 BC) 1. Fundamental logical equivalences Theorem 1. Let P , Q , and R be statements. 1. Commutative laws 1a. P Q Q P . 1.b P Q Q P . 2. Associative laws 2a. P ( Q R ) ( P Q ) R . 2b. P ( Q R ) ( P Q ) R . 3. Distributive laws 3a. P ( Q R ) ( P Q ) ( P R ) . 3b. P ( Q R ) ( P Q ) ( P R ) . 4. DeMorgan’s laws 4a. ( P Q ) ( P ) ( Q ) . 4b. ( P Q ) ( P ) ( Q ) . 5. Double negation: ∼∼ P P . 6. Theorem 2.17: P Q ( P ) Q . Example 2. Prove ( P Q ) P ( Q ). Proof. ( P Q ) ≡∼ (( P ) Q ) Theorem 2.17 ≡∼∼ P ∧ ∼ Q DeMorgan’s law P ∧ ∼ Q Double negation . Basically, we can treat like an equals symbol. We can save time, instead of writing out long truth tables. 2. Quantification Given an open sentence P ( x ); x S
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