Boolean Algebra Axioms and Theorems
The equivalence
Axioms:
The equivalence is associative, symmetric and has unit of true.
Theorems:
The equivalence is reflexive
The disjunction
Axioms:
The disjunction is associative, symmetric and idempotent.
It d
istributes over equivalence: p
(q
r)
p
q
p
r
Theorems:
Distributes over self: p
(q
r)
(p
q)
(p
r)
Has zero true: p
true
true
Has unit false: p
false
p
The conjunction
Axioms:
Definition: p
q
(p
q
p
q)
Binding is same as
disjunction
Theorems:
The conjunction is associative, symmetric and idempotent.
Distributes over self: p
(q
r)
(p
q)
(p
r)
Has unit true: p
true
p
Has zero false: p
false
false
Conjunction distributes over disjunction p
(q
r)
(p
q)
(p
r)
Disjunction distributes over conjunction p
(q
r)
(p
q)
(p
r)
Implication
Axioms:
Definition: p
q
p
q
q
Binding is less than conjunction and disjunction and greater than equivalence
Theorems:
p
true
p
p
p
q
p
true
q
q
p
q
p
q
p
p
(q
r)
p
q
r
p
(p
q)
p
q
Transitivity: (p
q)
(q
r)
(p
r)
(p
q)
(q
r)
Mutual implication: (p
q)
(q
p)
p
q
The consequence
Axioms:
Definition: p
q
p
q
p
Theorems:
p
q
q
p
The negation
Axioms:
Has the highest
binding
Definition:
(p
q)
p
q
Law of excluded middle: p
p
Theorems:
p
q
p
q
Double negative:
p
p
The discrepancy
Axioms:
Low binding, same as equivalence
Discrepancy is associative and symmetric
Associates with equivalence p
(q
r)
(p
q)
r
The rule (p
q)
(p
q)
Other Theorems:
The conjunction and disjunction distribute over each other
Absorption laws:
x
(x
y)
x
x
(x
y)
x
p
q
p
q
q
p
q
p
q
Contrapositive: p
q
q
p
False is defined as
true
DeMorg
an’s laws:
p
q
(p
q)
p
q
(p
q)
Complement laws
p
(
p
q)
p
q
p
(
p
q)
p
q
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