Laws and Theorems of Boolean Algebra

Boolean algebra is governed by a set of laws and theorems that are used to manipulate and simplify Boolean expressions. These laws are similar to those of ordinary algebra but with some important differences.

1.4.1 Basic Postulates and Properties

Law/Theorem	AND Form	OR Form
Identity Law	$A\cdot 1=A$	$A+0=A$
Null/Annihilation Law	$A\cdot 0=0$	$A+1=1$
Idempotent Law	$A\cdot A=A$	$A+A=A$
Complement Law	A⋅Aˉ=0	A+Aˉ=1
Commutative Law	$AB=BA$	$A+B=B+A$
Associative Law	$(AB)C=A(BC)$	$(A+B)+C=A+(B+C)$
Distributive Law	$A(B+C)=AB+AC$	$A+BC=(A+B)(A+C)$
Absorption Law	$A(A+B)=A$	$A+AB=A$
De Morgan’s Theorem	AB=Aˉ+Bˉ	A+B​=AˉBˉ
Involution Law	Aˉˉ=A	Aˉˉ=A

1.4.2 Duality Principle

The duality principle is a powerful concept in Boolean algebra. It states that if a Boolean expression is valid, its dual is also valid. The dual of an expression is obtained by:

1. Interchanging the AND and OR operations.
2. Interchanging the identity elements 0 and 1.
3. Leaving the variables unchanged.

For example, the dual of the distributive law $A(B+C)=AB+AC$ is $A+BC=(A+B)(A+C)$. Notice how each law in the table above has a dual counterpart.

1.4.3 De Morgan’s Theorems

De Morgan’s theorems are particularly important for simplifying expressions and converting between different logic forms.

1. Theorem 1: The complement of a product of variables is equal to the sum of the complements of the variables. AB=Aˉ+Bˉ
2. Theorem 2: The complement of a sum of variables is equal to the product of the complements of the variables. A+B​=AˉBˉ

These theorems can be extended to any number of variables. For example, ABC=Aˉ+Bˉ+Cˉ.