hw1soln

hw1soln - ECE 152A Fall 2006 10/10/2006 University of...

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ECE 152A – Fall 2006 10/10/2006 Homework #1 - Solution Page 1 of 33 University of California, Santa Barbara Department of Electrical and Computer Engineering ECE 152A – Digital Design Principles Homework #1 Solution Problem #1: Demonstrate by means of truth tables the validity of the following identities: 1. DeMorgan’s theorem for three variables: (xyz)’ = x’ + y’ + z’ x y z xyz (xyz)’ x’ y’ z’ x’ + y’ + z’ 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 0 2. The second distributive law: x + yz = (x + y)(x + z) x y z yz x + yz (x + y) (x + z) (x + y)(x + z) 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1
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ECE 152A – Fall 2006 10/10/2006 Homework #1 - Solution Page 2 of 33 3. The consensus theorem: xy + x’z + yz = xy + x’z x y z xy x’z yz xy + x’z + yz xy + x’z 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 Problem #2: Simplify the following Boolean expressions to a minimum number of literals: 1. x’y’ + xy + x’y x’ ( y’ + y) + xy x’ + xy (x’ + x) ( x’ + y) = x’ + y 2. (x + y) (x + y’) (x + y) (x + y’) { multiply out } xx + xy + xy’ + yy’ x + xy + xy’ = x
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ECE 152A – Fall 2006 10/10/2006 Homework #1 - Solution Page 3 of 33 3. x’y + xy’ + xy + x’y’ (x’ + x)y + (x’ + x)y’ y + y’ = 1 4. x’ + xy + xz’ + xy’z’ x’ + xy + xz’ x’ + x( y + z’) = x’ + y + z’ { by A + A’B = A + B } 5. xy’ + y’z’ + x’z’ = xy’ + x’z’ { by consensus theorem } Problem #3: Simplify the following Boolean expressions to a minimum number of literals: 1. ABC + A’B + ABC’ AB(C + C’) + A’B AB + A’B (A + A’)B = B 2. x’yz + xz z ( x’y + x) = z ( y + x)
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ECE 152A – Fall 2006 10/10/2006 Homework #1 - Solution Page 4 of 33 3. (x + y)’ (x’ + y’) x’y’ ( x’ + y’ ) x’y’x’ + x’y’y’ = x’y’ 4. xy + x(wz + wz’) xy + x(w(z + z’)
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This note was uploaded on 04/30/2008 for the course ECE 152a taught by Professor Johnson during the Summer '07 term at UCSB.

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hw1soln - ECE 152A Fall 2006 10/10/2006 University of...

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