EE101-U3-BooleanAlgebra-Nazarian-Spring12

Of decoders x y z f x y z a b 0 0 0 0 0 0 0 0 0 0 0 1

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Unformatted text preview: 2012 7 Using Decoders to Implement Functions •  Given any logic function, it can be implemented with the superposition of decoders X Y Z F 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 B x y z Shahin Nazarian/EE101/Spring 2012 C Z A B C 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 x y z Y 1 A X 0 x y z 0 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 8 Using Decoders to Implement Functions •  Given any logic function, it can be implemented with the superposition of decoders X Y Z F 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 x y z Shahin Nazarian/EE101/Spring 2012 C A B C F 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 F Z 1 B Y 0 A x y z X 0 x y z 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 9 Using Decoders to Implement Functions •  Given any logic function, it can be implemented with the superposition of decoders X Y Z F 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 C 0 A B C F 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 1x 0y 0z F 1 Z 1 B 1 Y 0 A 0 1x 0y 0z X 0 1x 0y 0z 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 F(1,0,0) = 1 Shahin Nazarian/EE101/Spring 2012 10 Using Decoders to Implement Functions •  Given any logic function, it can be implemented with the superposition of decoders X Y Z F 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 C 0 A B C F 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0x 1y 0z F 1 Z 1 B 0 Y 0 A 1 0x 1y 0z X 0 0x 1y 0z 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 F(0,1,0) = 1 Shahin Nazarian/EE101/Spring 2012 11 Using Decoders to Implement Functions •  Given any logic function, it can be implemented with the superposition of decoders X Y Z F 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 C 0 A B C F 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0x 1y 1z F 0 Z 1 B 0 Y 0 A 0 0x 1y 1z X 0 0x 1y 1z 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 F(0,1,1) = 0 Shahin Nazarian/EE101/Spring 2012 12 Boolean Algebra •  A set of theorems to help us manipulate logical expressions/equations •  Axioms = Basis / assumptions used •  Theorems = manipulations that we can use Shahin Nazarian/EE101/Spring 2012 13 Duality 1’s 0’s • + Shahin Nazarian/EE101/Spring 2012 F = A+B F^0 = *A . *B 14 Duality (Cont.) 1+0 Original expression •  0•1 Dual Taking the “dual” of both sides of an equation yields a new equation X+1=1 Original equation •  X•0=0 Dual Summary: •  Logic expression and its dual are not equal •  For two equal expressions (i.e., an equation) their duals are also equal (i.e., a new equation) Shahin Nazarian/EE101/Spring 2012 15 Axioms (A1) (A2) (A3) (A4) (A5) X = 0 if X ≠ 1 If X = 0, then X’ = 1 0•0=0 1•1=1 1•0=0•1=0 Shahin Nazarian/EE101/Spring 2012 (A1’) (A2’) (A3’...
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This note was uploaded on 03/13/2013 for the course EE 101 taught by Professor Redekopp during the Spring '06 term at USC.

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