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Laws of Boolean Algebra Cheat Sheet by

Axioms and Laws of Boolean Algebra
algebra     boolean

Identities

0 + X = X
0 · X = 0
1 + X = 1
1 · X = X
X + X = X
X · X = X

Negation

X + ~X = 1
~0 = 1
~1 = 0
~~X = X
X · ~X = 0

Laws

Commun­ative Law
A · B = B · A
 
A + B = B + A
Associ­ative Law
A · (B · C) = (A · B) · C
 
A + (B + C) = (A + B) +C
Distri­butive Law
A · (B + C) = A·B + A·C
 
A + B · C = (A+B)(A+C)
 

De Morgan's Laws

~(X · Y)
= ~X + ~Y
~(X + Y)
= ~X · ~Y
~(X · Y · Z)
= ~X + ~Y + ~Z
~(X + Y + Z)
= ~X · ~Y · ~Z

Theorems

Theorem 1
X + X · Y = X
Theorem 2
X + ~X · Y = X + Y
Theorem 3
X · Y + ~X · Z + Y · Z = X · Y + ~X · Z
Theorem 4
X(X + Y) = X
Theorem 5
X(~X + Y) = X · Y
Theorem 6
(X + Y)(X + ~Y) = X
Theorem 7
(X + Y)(~X + Z) = X · Z + ~X · Y
Theorem 8
(X + Y)(~X + Z)(Y + Z) = (X + Y)(~X + Z)

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