Laws and Theorems of Boolean Algebra

1a. X • 0 = 0 1b. X + 1 = 1 Annulment Law
2a. X • 1 = X 2b. X + 0 = X Identity Law
3a. X • X = X 3b. X + X = X Idempotent Law
4a. X • X = 0 4b. X + X = 1 Complement Law
5. X = X Double Negation Law
6a. X • Y = Y • X 6b. X + Y = Y + X Commutative Law
7a. X (Y Z) = (X Y) Z = (X Z) Y = X Y Z Associative Law
7b. X + (Y + Z) = (X + Y) + Z = (X + Z) + Y = X + Y + Z Associative Law
8a. X • (Y + Z) = X Y + X Z 8b. X + Y Z = (X + Y) • (X + Z) Distributive Law
9a. X • Y = X + Y 9b. X + Y = XY de Morgan's Theorem
10a. X • (X + Y) = X 10b. X + X Y = X Absorption Law
11a. (X + Y) • (X + Y) = X 11b. X Y + X Y = X Redundancy Law
12a. (X + Y) • Y = XY 12b. X Y + Y = X + Y Redundancy Law
13a. (X + Y) • (X + Z) • (Y + Z) = (X + Y) • (X + Z) Consensus Law
13b. X Y + X Z + Y Z = X Y + X Z Consensus Law
14a. X ⊕ Y = (X + Y) • (X + Y) 14b. X ⊕ Y = X Y + X Y XOR Gate
15a. X ⊙ Y = (X + Y) • (X • Y) 15b. X ⊙ Y = X Y + X Y XNOR Gate
15c. X ⊙ Y = (X + Y) • (X + Y) XNOR Gate

Gates

Standard DeMorgan's
NAND X = A • B X = A + B
NAND Gate
              
AND X = A • B X = A + B
AND Gate
NOR X = A + B X = AB
NOR Gate
              
OR X = A + B X = AB
OR Gate