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Discussion #10 1/16 Discussion #10 Logical Equivalences
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Discussion #10 2/16 Topics Laws Duals Manipulations / simplifications Normal forms –Definitions –Algebraic manipulation –Converting truth functions to logic expressions
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Discussion #10 3/16 Laws of , , and Excluded middle law Contradiction law P P T P P F NameLaw 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
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Discussion #10 4/16 Commutative laws P Q Q P P Q Q P NameLaw 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
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Discussion #10 5/16 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 FF TF FT TT QQ PP (P Q) QP De Morgan’s law holds.
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Discussion #10 6/16 Absorption Laws Prove algebraically … P (P Q) (P T) (P Q)identity P (T Q)distributive (factor) P Tdomination Pidentity P (P Q) P P (P Q) P Venn diagram proof … P Q
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Discussion #10 7/16 Duals To create the dual of a logical expression 1) swap propositional constants T and F, and 2) swap connective operators and . P P TExcluded Middle P P FContradiction The dual of a law is always a law! Thus, most laws come in pairs pairs of duals.
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Discussion #10 8/16 Why Duals of Laws are Always Laws Start with law P P T Negate both sides (P P) T Apply De Morgan’s law P P T Simplify negations P P F Since a law is a tautology, ( P ) ( P ) F substitute X for X Simplify negations P P F We can always do the following:
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Discussion #10 9/16 Normal Forms Normal forms are standard forms, sometimes called canonical or accepted forms. A logical expression is said to be in disjunctive normal form (DNF) if it is written as a disjunction, in which all terms are conjunctions of literals. Similarly, a logical expression is said to be in conjunctive normal form (CNF) if it is written as a conjunction of disjunctions of literals.
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Discussion #10 10/16 Disjunctive Normal Form (DNF) (.. .. .. ) (.. .. .. ) … (.. .. ) Term Literal, i.e. P or P Conjunctive Normal Form (CNF) (.. .. .. ) (.. .. .. ) … (.. .. ) Examples:(P Q) (P Q) P (Q R) DNF and CNF Examples:(P Q) (P Q) P (Q R)
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Discussion #10 11/16 Converting Expressions to DNF or CNF The following procedure converts an expression to DNF or CNF: 1.Remove all and . 2.Move inside. (Use De Morgan’s law.) 3.Use distributive laws to get proper form. Simplify as you go. (e.g. double-neg., idemp., comm., assoc.)
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Discussion #10 12/16 CNF Conversion Example (.. .. .. ) (.. .. .. ) … (.. .. ) (( P Q) R (P Q)) (( P Q) R ( P Q)) impl. ( P Q) R ( P Q) deM. ( P Q) R ( P Q) deM. (P Q) R (P Q) double neg. ((P R) ( Q R)) (P Q) distr. ((P R) (P Q)) distr. (( Q R) (P Q)) (((P R) P) ((P R) Q)) distr. ((( Q R) P) (( Q R) Q)) (P R) (P R Q) ( Q R) assoc. comm. idemp. (DNF)
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Discussion #10 13/16 CNF Conversion Example (.. .. .. ) (.. .. .. ) … (.. .. ) (( P Q) R (P Q)) (( P Q) R ( P Q)) impl. ( P Q) R ( P Q) deM. ( P Q) R ( P Q) deM. (P Q) R (P Q) double neg. ((P R) ( Q R)) (P Q) distr. ((P R) (P Q)) distr. (( Q R) (P Q)) (((P R) P) ((P R) Q)) distr. ((( Q R) P) (( Q R) Q)) (P R) (P R Q) ( Q R) assoc. comm. idemp. (DNF) CNF Using the commutative and idempotent laws on the previous step and then the distributive law, we obtain this formula as the conjunctive normal form.
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Discussion #10 14/16 CNF Conversion Example (.. .. .. ) (.. .. .. ) … (.. .. ) (( P Q) R (P Q)) (( P Q) R ( P Q)) impl. ( P Q) R ( P Q) deM. ( P Q) R ( P Q) deM. (P Q) R (P Q) double neg. ((P R) ( Q R)) (P Q) distr. ((P R) (P Q)) distr. (( Q R) (P Q)) (((P R) P) ((P R) Q)) distr. ((( Q R) P) (( Q R) Q)) (P R) (P R Q) ( Q R) assoc. comm. idemp. (DNF) (P R) (P R Q) ( Q R) (P R) (P R Q) (F Q R) - ident. (P R) ((P F) ( Q R)) - comm., distr. (P R) (F ( Q R)) - dominat. (P R) ( Q R) - ident.
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Discussion #10 15/16 DNF Expression Generation F T F F F T T F FFF TFF FTF TTF FFT TFT FTT TTT RQP (P Q R) (P Q R) ( P Q R) (P Q R) (P Q R) ( P Q R) minterms The only definition of is the truth table
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Discussion #10 16/16 CNF Expression Generation 1.Find . 2.Find the DNF of . 3.Then, use De Morgan’s law to get the CNF of (i.e. ( ) ) T F T F FFF TTF FFT TTT QP (P Q) ( P Q) ( P Q) (P Q) (P Q) ( P Q) DNF of f ((P Q) ( P Q)) (P Q) ( P Q) De Morgan’s ( P Q) (P Q) De Morgan’s, double neg. max terms } Form a conjunction of max terms
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