10-LogicalEquiv

10-LogicalEquiv - Discussion#10 Chapter 1 Section 5 1/19...

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Unformatted text preview: Discussion #10 Chapter 1, Section 5 1/19 Discussion #10 Logical Equivalences Discussion #10 Chapter 1, Section 5 2/19 Topics • Laws • Duals • Manipulations / simplifications • Normal forms – Definitions – Algebraic manipulation – Converting truth functions to logic expressions Discussion #10 Chapter 1, Section 5 3/19 Laws of ∧ , ∨ , and ¬ Excluded middle law Contradiction law P ∨ ¬ P ≡ T P ∧ ¬ P ≡ F Name Law Identity laws P ∨ F ≡ P P ∧ T ≡ P Domination laws P ∨ T ≡ T P ∧ F ≡ F Idempotent laws P ∨ P ≡ P P ∧ P ≡ P Double-negation law ¬ ( ¬ P) ≡ P Discussion #10 Chapter 1, Section 5 4/19 Commutative laws P ∨ Q ≡ Q ∨ P P ∧ Q ≡ Q ∧ P Name Law Associative laws (P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R) (P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R) Distributive laws (P ∨ Q) ∧ (P ∨ R) ≡ P ∨ (Q ∧ R) (P ∧ Q) ∨ (P ∧ R) ≡ P ∧ (Q ∨ R) De Morgan’s laws ¬ (P ∧ Q) ≡ ¬ P ∨ ¬ Q ¬ (P ∨ Q) ≡ ¬ P ∧ ¬ Q Absorption laws P ∨ (P ∧ Q) ≡ P P ∧ (P ∨ Q) ≡ P Discussion #10 Chapter 1, Section 5 5/19 Can prove all laws by truth tables… T F T F T T T F T T F F T T T T F F F T T T T F F F T F F T T T ¬ Q ∨ ¬ P ⇔ (P ∧ Q) ¬ Q P De Morgan’s law holds. Discussion #10 Chapter 1, Section 5 6/19 Absorption Laws Prove algebraically … P ∨ (P ∧ Q) ≡ (P ∧ T) ∨ (P ∧ Q) identity ≡ P ∧ (T ∨ Q) distributive (factor) ≡ P ∧ T domination ≡ P identity P ∨ (P ∧ Q) ≡ P P ∧ (P ∨ Q) ≡ P Venn diagram proof … P Q Discussion #10 Chapter 1, Section 5 7/19 Duals • To create the dual of a logical expression 1) swap propositional constants T and F, and 2) swap connective operators ∧ and ∨ ....
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10-LogicalEquiv - Discussion#10 Chapter 1 Section 5 1/19...

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