Lecture08EquationalProofs

Lecture08EquationalProofs - Lecture 8 CS 2603 Applied Logic...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 2
… 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 17

Lecture08EquationalProofs - Lecture 8 CS 2603 Applied Logic...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online