EE101-U3-BooleanAlgebra-Nazarian-Spring12

Specific input combination and outputs 0 when found

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Unformatted text preview: EE101/Spring 2012 M0 M1 M2 M3 M4 M5 M6 M7 X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 F 0 0 1 1 0 1 0 1 38 Example Shahin Nazarian/EE101/Spring 2012 39 T.T. => Circuit w minterm ? W=m1 W=Pi(m0,m2,m3,m4, m5,m6,m7 maxterm Shahin Nazarian/EE101/Spring 2012 40 Unique Representations not minterm x 0 0 0 0 1 1 1 1 Truth Table yz 00 01 10 11 00 01 10 11 Canonical Sum P 0 0 1 1 0 1 0 1 rows where P=1 Yields SOP equation, ANDOR circuit Shahin Nazarian/EE101/Spring 2012 maxterm Canonical Product rows where P=0 Yields POS equation, ORAND circuit 41 T.T. & Canonical Representations Shahin Nazarian/EE101/Spring 2012 42 Digital Design Goals area Speed fastest (in descending order) power Shahin Nazarian/EE101/Spring 2012 difficult to create positive logic, difficult to create negative gates without using inverters. 43 2 & 3 Variable Theorems T6 X+Y = Y+X T6’ XY = YX T7 (X+Y)+Z = X+(Y +Z) T7’ (XY)Z = X(YZ) T8 XY+XZ = X(Y+Z) T8’ (X+Y)(X+Z) = X+YZ T9 X + XY = X T9’ X(X+Y) = X T10 XY + XY’ = X T10’ (X+Y)(X+Y’) = X T11 XY + X’Z + YZ = XY + X’Z T11’ (X+Y)(X’+Z)(Y+Z) = (X+Y)(X’+Z) Shahin Nazarian/EE101/Spring 2012 44 Covering Theorems – Proof Shahin Nazarian/EE101/Spring 2012 45 Venn Diagrams •  Graphical method of expressing a logic function of 1-, 2-, or 3-variables •  Start with a box •  Add circles for each variable of the function x=0 y=0 x=1, y=0 x Shahin Nazarian/EE101/Spring 2012 x=1 y=1 x=0, y=1 y 46 Single Variable Venn Diagram x Shahin Nazarian/EE101/Spring 2012 47 Single Variable Venn Diagram •  Shade the areas where a function is true •  Area inside the circle represents where the variable is true This area is where x = 1 x F=x Shahin Nazarian/EE101/Spring 2012 48 Single Variable Venn Diagram •  Area outside the circle represents where the variable is false This is where x = 0 x F = x’ Shahin Nazarian/EE101/Spring 2012 49 2-Variable Venn Diagrams •  Add another circle for y •  Each disjoint region represents a distinct combination of inputs (i.e. a minterm or maxterm) x Shahin Nazarian/EE101/Spring 2012 y 50 2-Variable Venn Diagrams •  Add another circle for y •  Each disjoint region represents a distinct combination of inputs (i.e. a minterm or maxterm) x=1,y=1 x=1,y=0 x=0,y=1 x y x=0,y=0 Shahin Nazarian/EE101/Spring 2012 51 2-Variable Venn Diagrams F = x+y x y F = x•y x Shahin Nazarian/EE101/Spring 2012 y 52 F=X’•Y (AND’ing Venn Diagrams) x’ x y y x y F = x’ • y x Shahin Nazarian/EE101/Spring 2012 y 53 Venn Diagrams (OR’ing) x x y y x y F = x+y x Shahin Nazarian/EE101/Spring 2012 y 54 Venn Diagrams Practice F = x’+y x’ x y y x y F = x’+y x Shahin Nazarian/EE101...
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