HW2_Sol

# HW2_Sol - ECE 15A Homework 2 Solutions 1 Write out the...

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ECE 15A Homework 2 Solutions 1. Write out the proof that a(a+b) = a in theorem 4, referring each step to the correct postulate: a(a+b) = a aa + ab by Distributive Law a + ab by Theorem 2 a1+ab P2 a(1+b) by Distributive Law a=1a T2 2. Prove that in every Boolean Algebra every triple of elements a,b,c satisfies the identity ab+bc+ca=(a+b)(b+c)(c+a) ab+bc+ca b(a+c)+ca Distributive Law (b+ac)(a+c+ac) Distributive Law (b+a)(b+c)(a+a+c)(a+c+c) Distributive Law (b+a)(b+c)(a+c) 3. Prove, that if a + x = b + x and a + x’ = b + x’ , then a = b . a = a + ax + ax’ Thm 4 b = b + bx + bx’ Thm 4

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a = a + ax + ax’ = aa + ax + ax’ + xx’ P2, P4 = a ( a + x) + x’( a + x) P3 = ( a + x ) ( a + x’) P3 = ( b + x ) ( b + x’) Given = bb + bx’ + xb + xx’ P3 = b + bx + bx’ P4 = b 4. Prove, that if ax = bx and ax’ = bx’ , then a = b . a = a ( x + x’) P4 = a x + a x P3 = b x + b x Given = b ( x + x ) P3 = b P 4 ==> a = b 5. Show that set {a,b,c,d} with operations (+) and (.) as defined is a Boolean algebra. To prove that the operations are a Boolean algebra, we have to show that the postulates P1- P4 (see lecture notes #3, slide 14) hold.
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## This note was uploaded on 01/13/2010 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.

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HW2_Sol - ECE 15A Homework 2 Solutions 1 Write out the...

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