Lecture09TheBigSurprise

Lecture09TheBigSurprise - Lecture 9- CS 2603 Applied Logic...

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 9—CS 2603 Applied Logic for Hardware and Software More Equations and The Big Surprise
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 Two Implications for the Price of One (p q) r = (p r) (q r) { imp} equations {rule} substitution [formula in eqn / variable in rule] (p q) r = ( ¬ (p q)) r {imp} [p q/a] ±[r/b] = (( ¬ p) ( ¬ q)) r {DeMorgan } [p /a] [q /b] = r (( ¬ p) ( ¬ q)) { comm} [( ¬ p) ( ¬ q) /a] [r /b] = ( r ( ¬ p)) ( r ( ¬ q)) { dist/ } [r /a] [ ¬ p /b] [ ¬ q /c] = ( ( ¬ p) r) ( r ( ¬ q)) { comm} [r /a] [ ¬ p/b] = ( ( ¬ p) r) ( ( ¬ q) r) { comm} [r /a] [ ¬ q/b] = (p r) ( ( ¬ q) r) {imp} [p /a] [r /b] = (p r) (q r) {imp} [q /a] [r /b] QED
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 True, Incognito True = p (( ¬ p) q) equations {rule} substitution p (( ¬ p) q) =( ¬ p) (( ¬ p) q) {imp} [p /a] [( ¬ p) q /b] ¬ p) (( ¬ ( ¬ p)) q) {imp} [ ¬ p /a] [q /b] ¬ p) (p q) {dbl neg} [p /a] =(( ¬ p) p) q { assoc} [ ¬ p /a] [p /b] [q /c] =(p ( ¬ p)) q { comm} [ ¬ p /a] [p /b] =True q { comp} [p /a] =q True { comm} [True /a] [q /b] { null} [q /a] QED
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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 True = p (( ¬ p) q) a different proof equations {rule} substitution p (( ¬ p) q) =(p ( ¬ p)) q {currying} [p /a] [( ¬ p)/b] [q/c] =Fa lse q { complement} [p /a] =( ¬ False ) q {implication} [p /a] [q / b] =True q {negate False} =q True { comm} [q /a] [True /b] { null} [q /a] QED
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Lecture09TheBigSurprise - Lecture 9- CS 2603 Applied Logic...

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