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EE101-U3-BooleanAlgebra-Nazarian-Spring12

# Yw f xy z yw shahin nazarianee101spring 2012 67

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Unformatted text preview: + ZY+ZZ + XZ +XY 0 = X+Y + YZ+ XZ +XY = X+Y(1 + Z + X) + XZ 1 = X+Y + XZ (T8 and T2) = X(Z+Z) +Y + XZ (T1’ and T5) = XZ+XZ +Y + XZ (T8) = XZ+XZ + XZ +Y + XZ (T3) = X(Z+Z) + Z(X+X) +Y This doesn’t look like it can be simplified further but it can (T8 and T8’) (T8) = X + Z +Y Shahin Nazarian/EE101/Spring 2012 (T5’ and T1) (T8 and T2) (T5 and T1’) 70 Simplification – Example Shahin Nazarian/EE101/Spring 2012 71 Simplification – Examples Shahin Nazarian/EE101/Spring 2012 72 Simplification – Schematic Example Shahin Nazarian/EE101/Spring 2012 73 Simplification – Example •  The police are looking for drunk drivers. They can arrest a person, if his breath test is positive OR if they find an open container in the car AND the contents are alcoholic Arrest = Breath + Open_Container • Contents_Alcoholic •  Now let us use the + over • distribution and rewrite it as: Arrest = (Breath + Open_Container) • (Breath + Contents_Alcoholic) Shahin Nazarian/EE101/Spring 2012 74 DeMorgan’s Logic Equivalence Shahin Nazarian/EE101/Spring 2012 75 Negative Logic •  Recall it is up to us humans to assign meaning to the two voltage levels •  Thus, far we’ve used (unknowingly) the positive logic convention where 1 means true and 0 means false •  In negative logic 0 means true and 1 means false volts 1=true/on (Value/Meaning) 0=false/off volts 1=false/off (Value/Meaning) 0=true/on time Positive Logic Convention Shahin Nazarian/EE101/Spring 2012 Negative Logic Convention time 76 Negative Logic ‘AND’ Function Traditional P.L. AND N.L. AND function X 0 0 1 1 X 1 1 0 0 Y 0 1 0 1 Z 0 0 0 1 Traditional AND gate functionality assumes positive logic convention Y 1 0 1 0 Z 1 1 1 0 Given negative logic signals, we can invert to positive logic, perform the AND operation, then convert back to negative logic Shahin Nazarian/EE101/Spring 2012 N.L. AND = P.L. OR X 0 0 1 1 Y 0 1 0 1 Z 0 1 1 1 However, we then see that an OR gate implements the negative logic ‘AND’ function 77 Negative Logic ‘OR’ Function Traditional P.L. OR X 0 0 1 1 Y 0 1 0 1 N.L. OR function Z 0 1 1 1 Traditional OR gate functionality assumes positive logic convention X 1 1 0 0 Y 1 0 1 0 Z 1 0 0 0 Given negative logic signals, we can invert to positive logic, perform the OR operation, then convert back to negative logic Shahin Nazarian/EE101/Spring 2012 N.L. OR = P.L. AND X 0 0 1 1 Y 0 1 0 1 Z 0 0 0 1 However, we then...
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