Boolean Algebra Axioms and Theorems

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