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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:
The commutative property for a logical
OR, is:
Using electronic gates, the distributive property is:
The De Morgan's laws
can transform logical ORs into logical ANDs (negations are necessary)
and can electronically be described this way:
or
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'Axioms' to home
From
'Axioms' to 'Boolean Algebra'


