A few basic properties of Boolean Algebra are summarized in the following table. Each relationship in the table has both AND and OR form as a result of the principle of duality. The dual form is obtained by exchanging AND's and OR's, and exchanging 1 and 0.
| Relationship | Dual | Property |
|---|---|---|
| A & B = B & A
A & (B V C) = (A & B) V (A & C) 1 & A =A A & not(A) = 0 0 & A = 0 A & A = A A & (B & C) = (A & B) & C not(not(A)) = A not(A & B) = not(A) V not(B) A & (A V B) = A |
A V B = B V A
A V (B&C) = (A V B)&(A V C) 0 V A = A A V not(A) = 1 1 V A = 1 A V A =A A V (B V C) = (A V B) V C not(A V B) = not(A) & not(B) A V (A & B) = A |
Commutative
Distributive Identity Complement Zero and one theorems Idempotence Associative Involution DeMorgan's theorem Absorption theorem |
Some of these relationships, e.g. the commutative or distributive properties, or DeMorgan's theorem can be extended for more than two variables.
Here is a proof for DeMorgan's theorem for the two-variable case.
| A B | not(A&B) | not(A) V not(B) | not(A V B) | not(A) & not(B) |
|---|---|---|---|---|
| 0 0 0 1 1 0 1 1 |
1 1 1 0 |
1 1 1 0 |
1 0 0 0 |
1 0 0 0 |