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LOGIC3-2_r2

LOGIC3-2_r2 - Chapter 3-2 Algebraic Simplification...

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Microwave & Communication Lab. Chapter 3-2 Algebraic Simplification

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Microwave & Communication Lab. proof ： XY+X’Z+YZ = XY+X’Z+(X+X’)YZ = XY+X’Z+XYZ+X’YZ = XY(1+Z)+X’Z(1+Y) = XY+X’Z § The Consensus Theorem
Microwave & Communication Lab. EX： a’b’+ac+bc’+b’c+ab Sol： Let x = a’ , y = b’ , z = c a’b’+ac+b’c = a’b’+ac b’c is consensus term so, a’b’+ac+bc’+b’c+ab = a’b’+ac+bc’+ab Let x=c , x’ = c’ , y = a , z = b ca+c’b+ab = ca+c’b ab is consensus term so , a’b’+ac+bc’+ab = a’b’+ac+bc’ hence, a’b’+ac+bc’+b’c+ab=a’b’+ac+bc’

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Microwave & Communication Lab. § The dual form of the consensus theorem ： (X+Y)(X’+Z)(Y+Z) = (X+Y)(X’+Z) i.e. , If F = G F D =G D proof： Let F = XY+X’Z+YZ , G = XY+X’Z F D = (X+Y)(X’+Z)(Y+Z) G D = (X+Y)(X’+Z) F D = (X+Y)(X’+Z)(Y+Z) = (XZ+X’Y)(Y+Z) = XYZ+XZ+X’Y+X’YZ = XZ+X’Y+YZ = XZ+X’Y+YZ+XX’ = X(X’+Z)+Y(X’+Z) = (X+Y)(X’+Z) = G D
Microwave & Communication Lab. EX Simplify (a+b+c’)(a+b+d’)(b+c+d’) Sol Let x = c, x’ = c, z = a+b, y = b +d’, y+z = a+b+d’

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LOGIC3-2_r2 - Chapter 3-2 Algebraic Simplification...

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