EE101-U3-BooleanAlgebra-Nazarian-Spring12

E a minterm or maxterm x shahin nazarianee101spring

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Unformatted text preview: /Spring 2012 y 55 Venn Diagrams Practice F = (x•y)’ x x Shahin Nazarian/EE101/Spring 2012 y y 56 Venn Diagrams Answer F = (x•y)’ x•y x y F = (x•y)’ x Shahin Nazarian/EE101/Spring 2012 y 57 3-Variable Venn Diagrams •  All the same rules apply •  Since there are 3-variables, we must have 8 disjoint regions (1 for each combination / minterm / maxterm) x y z Shahin Nazarian/EE101/Spring 2012 58 3-Variable Venn Diagrams •  Each area represents a different combination of x, y, and z This area represent where x=1 and y=1 and z=0 (i.e. F=x•y•z’) x y z Shahin Nazarian/EE101/Spring 2012 59 3-Variable Venn Diagrams •  Each area represents a different combination of x, y, and z This area represent where x=1 and y=1 and z=0 (i.e. F=x•y•z’) This area represent where x=1 and y=1 and z=1 (i.e. F=x•y•z) x y z Shahin Nazarian/EE101/Spring 2012 60 3-Variable Venn Diagrams •  Each area represents a different combination of x, y, and z This area represent where x=1 and y=1 and z=0 (i.e. F=x•y•z’) This area represent where x=1 and y=1 and z=1 (i.e. F=x•y•z) x y This area represent where x=0 and y=0 and z=1 (i.e. F=x’•y’•z) z Shahin Nazarian/EE101/Spring 2012 61 3-Variable Venn Diagrams F = x + y•z x x y•z y x z y z F = x + y•z x y z Shahin Nazarian/EE101/Spring 2012 62 T9 Proof x + xy = x x xy + x = x y y x x Shahin Nazarian/EE101/Spring 2012 y 63 T9’ Proof x(x+y)= x x x+y • x = x y y x x Shahin Nazarian/EE101/Spring 2012 y 64 DeMorgan’s Theorem •  Inverting output of an AND gate = inverting the inputs of an OR gate •  Inverting output of an OR gate = inverting the inputs of an AND gate A function’s inverse is equivalent to inverting all the inputs and changing AND to OR and vice versa A B Out A B Out 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 A B Out A B Out 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 A•B A+B Shahin Nazarian/EE101/Spring 2012 A+B A•B 65 DeMorgan’s Theorem F = (X+Y) + Z • (Y+W) F = (X+Y) + Z • (Y+W) F = (X+Y) • (Z • (Y+W)) F = (X•Y) • (Z + (Y+W)) F = (X•Y) • (Z + (Y•W)) Shahin Nazarian/EE101/Spring 2012 66 Generalized DeMorgan’s Theorem F = (X+Y) + Z • (Y+W) F = X+Y + (Z • (Y+W)) F = X•Y • (Z + (Y•W)) Shahin Nazarian/EE101/Spring 2012 67 Example Shahin Nazarian/EE101/Spring 2012 68 Logic Simplification •  Tips •  Single variables in T8-T11 can represent entire sub-expressions in an equation –  If you can find something of the form of T8T11 you can apply that theorem •  Use T8 to distribute and get SOP or POS form •  Factor common subexpressions (use T8), then see if anything reduces •  Apply DeMorgan’s to get only literals •  Replicate terms if needed Shahin Nazarian/EE101/Spring 2012 69 Simplification – Example 1 F = (X+Y) + Z(Y+Z) + X(Z+Y) = X+Y...
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