Lecture08EquationalProofs

# Lecture08EquationalProofs - Lecture 8 CS 2603 Applied Logic...

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Lecture 8 —CS 2603 Applied Logic for Hardware and Software Reasoning with Equations Back to the Future: High-School Algebra

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 Some Algebraic Equations ± a + 0 = a {+ identity} ± (-a) + a = 0 {+ complement} ± a × 1 = a { × identity} ± a × 0 = 0 { × null} ± a + b = b + a {+ commutative} ± a + (b+c) = (a+b) + c {+ associative} ± a × (b+c) = a × b+ ±a × c {distributive law} Equations go both ways
… and then a miracle happened … = 1 Negative × Negative = Positive (-1) × (-1) = 1 proof by equational reasoning (-1) × (-1) = ((-1) × (-1)) + 0 {+ id} = ((-1) × (-1)) + ((-1) + 1) {+ comp} = ( ((-1) × (-1)) + (-1) ) + 1 {+ assoc} = (((-1) × (-1)) + (-1) × 1 ) + 1 { × id} = ( (-1) × ((-1) + 1)) + 1 {dist law} = ((-1) × 0 ) + 1 {+ comp} = 0 + 1 { × null} = 1 + 0 {+ comm} = 1 {+ id} QED CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 Some Basic Equations of Boolean Algebra page 1 From Fig 2.1, Hall & O’Donnell, Di screte Math with a C om puter , Spr inger , 2000
CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 5 More Basic Equations of Boolean Algebra page 2 From Fig 2.1, Hall & O’Donnell, Di screte Math with a C om puter , Spr inger , 2000

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CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 6 Deriving a New Equation (p False) (q True) = q equations {rule} substitution [formula in eqn / variable in rule] (p False) (q True) = False (q True) { null} [p /a] = (q True) False { comm} [False /a] [q True /b] = q True { id} [q True /a] = q { id} [q /a] QED
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Lecture08EquationalProofs - Lecture 8 CS 2603 Applied Logic...

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