Lecture07EquationalProofs

Lecture07EquationalProofs - Lecture 7 CS 1813 Discrete...

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CS 1813 Discrete 1 Lecture 7 CS 1813 – Discrete Mathematics Equational Reasoning Back to the Future: High-School Algebra
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CS 1813 Discrete 2 Some Laws of Algebra a + 0 = a {+ identity} (-a) + a = 0 {+ complement} ×  1 = a { ×  identity} ×  0 = 0 { ×  null} a + b = b + a {+ commutative} a + (b+c) = (a+b) + c {+ associative} a × (b+c) = a × b + a × {distributive law} Equations go both ways
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CS 1813 Discrete 3 Theorem (-1) × (-1) = 1    (-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 proof by equational reasoning
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CS 1813 Discrete 4 Laws of Boolean Algebra page 1 From Fig 2.1, Hall & O’Donnell, Discrete Math with a Computer , Springer, 2000
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Lecture07EquationalProofs - Lecture 7 CS 1813 Discrete...

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