HW2_soln

HW2_soln - ECE 15A Homework 2 Solutions 1. Write out the...

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ECE 15A Homework 2 Solutions 1. Write out the proof that a0 = 0 in theorem 3, referring each step to the correct postulate: 0 = aa' P4 = a a . 1 P2 = a (a’.1) P1 = a (a’. (a + a’)) P4 = a (a’a + a’a’) P3 = a ( 0 + a’) P4 = a0 + aa’ P3 = a0 + 0 P4 = a 0 P2 ==> a0 = 0 2. Prove that in every Boolean Algebra a + a’b = a + b for every pair of elements a and b. a + b = a + a’b = ( a + a’)(a + b) Distributive property = 1 . ( a + b ) P4 = a + b P2 ==> a + b = a + a’b
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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 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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P1: The operations “+” and “.” are commutative.
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HW2_soln - ECE 15A Homework 2 Solutions 1. Write out the...

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