PPT Slide
Axioms and theorems of Boolean algebra (cont'd)
Useful axioms and theorems of Boolean algebra (continued):
- duality: 14. (X + Y + ...)D = X • Y • ... 15D. (X • Y • ...)D= X + Y + ...
- generalized duality: 15. fD(X1,X2,...,Xn,0,1,+,•) = f(X1,X2,...,Xn,1,0,•,+)
- multiplication and factoring: 16. (X + Y) • (X' + Z) = X • Z + X' • Y 16D. X • Y + X' • Z = (X + Z) • (X' + Y)
- concensus: 17. (X • Y) + (Y • Z) + (X' • Z) = 17D. (X + Y) • (Y + Z) • (X' + Z) = X • Y + X' • Z (X + Y) • (X' + Z)
Duality
- a dual of a Boolean expression is derived by replacing • by +, + by •, 0 by 1, and 1 by 0, and leaving variables unchanged
- any theorem that can be proven is thus also proven for its dual!
- a meta-theorem (a theorem about theorems)