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Axioms - Laws of Boolean Algebra

Boolean algebra is the algebra of propositions.
Propositions are denoted by letters, such as A, B, x or y, etc.

In the following axioms and theorems (laws of boolean algebra), the '+' or 'V' signs represent a logical OR (or conjunction), the '.' or '^' signs represent a logical AND (or disjunction), and 'Ź' or '~' represent a logical NOT ( or negation).

Every proposition has two possible values: 1 (or T) when the proposition is true and 0 (or F) when the proposition is false.

The negation of A is written as ŹA (or ~A) and read as 'not A'. If A is true then ŹA is false. Conversely, if A is false then ŹA is true.

Descript.	OR form	AND form	Other way to express it:
Axiom	x+0 = x	x.1 = x	A V F = A A ^ T = A
Commutative	x+y = y+x	x.y = y.x	A V B = B V A A ^ B = B ^ A
Distributive	x.(y+z) = (x.y)+(x.z)	x+y.z = (x+y).(x+z)	A ^ (B V C) = (A ^ B) V (A ^ C) A V B ^ C = (A V B) ^ (A V C)
Axiom	x+Źx = 1	x.Źx = 0	A V ŹA = T A ^ ŹA = F
Theorem	x+x = x	x.x = x	A V A = A A ^ A = A
Theorem	x+1 = 1	x.0 = 0	A V T = T A ^ F = F
Theorem	ŹŹx = x		Ź(ŹA) = A
Associativity	x+(y+z) = (x+y)+z	x.(y.z) = (x.y).z	A V (B V C) = (A V B) V C A ^ (B ^ C) = (A ^ B) ^ C
Absorption	x+x.y = x	x.(x+y) = x	A V A ^ B = A A ^ (A V B) = A
DeMorgan's laws	x+y = Ź(Źx.Źy)	x.y = Ź(Źx+Źy)	A V B = Ź(ŹA ^ ŹB) A ^ B = Ź(ŹA V ŹB)

Using logical gates, the commutative property for a logical AND is:

The commutative property for a logical OR, is:

Using electronic gates, the distributive property is:

distributive property using gates

The De Morgan's laws can transform logical ORs into logical ANDs (negations are necessary) and can electronically be described this way:

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