# Boolean Algebra Axioms and Theorems

Electronics - Devices

Oct 10, 2013 (4 years and 7 months ago)

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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