Digital Lecture 7

# Digital Lecture 7 - 17-Jan-067:15 PM Boolean Algebra EEL...

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17-Jan-06—7:15 PM 1 1 University of Florida, EEL 3701 – File 07 © Drs. Schwartz & Arroyo Boolean Algebra EEL 3701 1 University of Florida, EEL 3701 – File 07 © Drs. Schwartz & Arroyo EEL 3701 Menu • Boolean Algebra >Theorems >Definitions Look into my . .. See examples on web: BooleanAlgebra.PDF EEL 3701 2 University of Florida, EEL 3701 – File 07 © Drs. Schwartz & Arroyo EEL 3701 The Mathematics of Logic Design - Boolean Algebra Basic Postulates & Theorems > Identity Laws 1. X + 0 = X X • 1 = X 2. X + 1 = 1 X • 0 = 0 > Indempotent Laws 3. X + X = X X • X = X > Involution & Complementarity Laws 4. (X’)’ = X 5. X + X’ = 1 X • X’ = 0 > Commutative Laws 6. X + Y = Y + X X • Y = Y • X > Associative Laws 7. (X+Y)+Z=X+(Y+Z)=X+Y+Z (XY)•Z=X•(YZ)=XYZ > Absorption Laws 8. X • (X +Y) = X X + (X•Y) = X

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17-Jan-06—7:15 PM 2 2 University of Florida, EEL 3701 – File 07 © Drs. Schwartz & Arroyo Boolean Algebra EEL 3701 3 University of Florida, EEL 3701 – File 07 © Drs. Schwartz & Arroyo EEL 3701 The Mathematics of Logic Design - Boolean Algebra Important Theorems > Distributive Laws 9. X•(Y+Z) = X•Y+X•Z X+(Y•Z)=(X+Y)•(X+Z) > De Morgan’s Law 10. (X+Y+Z)’ = X’•Y’•Z’ (X•Y•Z)’ = X’+Y’ +Z’ > Duality If we treat + and • as dual pairs, and 0 and 1 as dual pairs, the theorems in one column can be deduced from the other, e.g., 1. X + 0 = X 2. X + 1 = 1 5. X + X’ = 1 1D. X • 1 = X 2D. X • 0 = 0 5D. X • X’ = 0 EEL 3701 4 University of Florida, EEL 3701 – File 07 © Drs. Schwartz & Arroyo EEL 3701 Truth Table Proof of DeMorgan’s Law (and use of Duality) /(A + B) = /A • /B /(A • B) = /A + /B Dual A B /(A+B) /A•/B /(A•B) /A + /B 00 1 1 1 1 01 0 0 1 1 10 0 0 1 1 11 0 0 0 0 QED
17-Jan-06—7:15 PM 3 3 University of Florida, EEL 3701 – File 07 © Drs. Schwartz & Arroyo

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## Digital Lecture 7 - 17-Jan-067:15 PM Boolean Algebra EEL...

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