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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 '¬' 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 ¬A (or ~A) and read as 'not A'. If A is true then ¬A is false. Conversely, if A is false then ¬A is true.

Basic Laws of Boolean Algebra

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+¬x = 1 x.¬x = 0 A V ¬A = T
A ^ ¬A = 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 ¬¬x = x   ¬(¬A) = 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 = ¬(¬x.¬y) x.y = ¬(¬x+¬y) A V B = ¬(¬A ^ ¬B)
A ^ B = ¬(¬A V ¬B)

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

commutativity 1

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:

De Morgan's law 1

De Morgan law 2

 From 'Axioms' to home

 From 'Axioms' to 'Boolean Algebra'

AND gate

OR gate

NOT gate

XOR gate

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