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)
DeMorgans theorem:
16. (X + Y + Z + ...) = XYZ...
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 + XZ + YZ = XY + XZ
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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