LAWS AND THEOREMS OF BOOLEAN ALGEBRA

 Identity Dual Operations with 0 and 1: 1. X + 0 = X (identity) 3. X + 1 = 1 (null element) 2. X.1 = X 4. X.0 = 0 Idempotency theorem: 5. X + X = X 6. X.X = X Complementarity: 7. X + X’ = 1 8. X.X’ = 0 Involution theorem: 9. (X’)’ = X Identities for multiple variables Cummutative law:  10. X + Y = Y + X 11. X.Y = Y X Associative law: 12. (X + Y) + Z = X + (Y + Z)      = X + Y + Z 13. (XY)Z = X(YZ)        = XYZ Distributive law: 14. X(Y + Z) = XY + XZ 15. X + (YZ) = (X + Y)(X + Z) DeMorgan’s theorem: 16. (X + Y + Z + ...)’ = X’Y’Z’... or {f(X1,X2,...,Xn,0,1,+,.)} = {f(X1’,X2’,...,Xn’,1,0,.,+)} 17. (XYZ...)’ = X’ + Y’ + Z’ + ... Simplification theorems: 18. XY + XY’ = X (uniting) 20. X + XY = X (absorption) 22. (X + Y’)Y = XY (adsorption) 19. (X + Y)(X + Y’) = X 21. X(X + Y) = X 23. XY’ + Y = X + Y Consensus theorem: 24. XY + X’Z + YZ = XY + X’Z 25. (X + Y)(X’ + Z)(Y + Z) = (X + Y)(X’ + Z) Duality: 26. (X + Y + Z + ...)D = XYZ... or {f(X1,X2,...,Xn,0,1,+,.)}D = f(X1,X2,...,Xn,1,0,.,+) 27. (XYZ ...)D = X + Y + Z + ...

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