hw2_soln - Homework 2 Solutions ECE 0132 September 20, 2005...

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Unformatted text preview: Homework 2 Solutions ECE 0132 September 20, 2005 Problem 1 a) x 1 ( x 2 x 3 ) = (x 1 x 2 ) x 3 b) x 1 x 2 __ = x 1 __ x 2 = x 1 x 2 ______ c) (x 1 x 2 ) (x 2 x 3 ) = x 1 x 3 x1 x2 x3 (x 1 x 2 ) (x 2 x 3 ) LHS RHS : x1 x3 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 RHS = LHS for all inputs d) (x 1 x 2 ______ ) (x 2 x 3 ______ ) = x 1 __ x 2 __ x 3 __ + x 1 x 2 x x1 x2 x3 (x 1 x 2 ______ ) (x 2 x 3 ______ ) LHS x 1 __ x 2 __ x 3 __ x 1 x 2 x 3 RHS 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 RHS = LHS for all inputs Problem 2: There are multiple ways to solve these problems. a) (a + b)(b + c)(a + c) = ab + ac + bc Property Distributivity: (ab + ac + bb + bc)(a+c) = Theorem 7a [x*x = x] : (ab + ac + b + bc)(a+c) = Property Distributivity: (aab + abc + aac + ab + bc + abc + bcc) = Theorem 7a [x*x = x] : (ab + abc + ac + ab + bc + abc + bc) = Theorem 7b [x+x = x] : (ab + abc +ac + bc) = Property Absorption 13a[x + xy = x]: ab + ac + bc = ab + ac + bc b) x 1 x 2 + x 1 x 2 __ + x 1 __ x 2 __ = x 1 + x 2 __ Theorem 7a [x*x = x] (used to replicate 2 nd term) : x 1 x 2 + x 1 x 2 __ + x 1 x 2 __ + x 1 __ x 2 __ = Property 14a Combining: x 1 + x 2 __ = x 1 + x 2 __ c) (x 1 __ x 3 __ + x 1 __ x 2 + x 1 __ x 3 + x 1 x 2 ) = x 1 x 2 __ Property DeMorgans 15a: ) 2 1 )( 3 1 )(...
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hw2_soln - Homework 2 Solutions ECE 0132 September 20, 2005...

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