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16-ValidEquiv

16-ValidEquiv - Discussion#16 Chapter 2 Section 2.4 1/12...

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Unformatted text preview: Discussion #16 Chapter 2, Section 2.4 1/12 Discussion #16 Validity & Equivalences Discussion #16 Chapter 2, Section 2.4 2/12 Topics • Validity – Tautologies with Interpretations – Contradictions with Interpretations • Logical Equivalences Involving Quantifiers • Rectification Discussion #16 Chapter 2, Section 2.4 3/12 Validity • An expression that is true for all interpretations is said to be valid . (A valid expression is also call a tautology.) • An expression that is true for no interpretation is said to be contradictory . (A contradictory expression is also called a contradiction.) • If A is valid, ¬ A is contradictory. (a tautology) (a contradiction) • Examples: – P(x, y) ⇒ P(x, y) ≡ ¬ P(x, y) ∨ P(x, y) is valid – P(x, y) ∧ ¬ P(x, y) is contradictory Discussion #16 Chapter 2, Section 2.4 4/12 Laws are Valid • All laws are valid. – de Morgan’s: ¬ (P(x) ∧ Q(y)) ≡ ¬ P(x) ∨ ¬ Q(y) – Identity: P(x) ∧ T ≡ P(x) • When we replace ≡ by ⇔ , the resulting expression is true for all interpretations....
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16-ValidEquiv - Discussion#16 Chapter 2 Section 2.4 1/12...

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