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

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

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