Canonical and Standard Form

Boolean algebras are represented in a logical way in canonical and standard form mainly for the simplification of variables. It is the algebraic form of simplification of boolean functions. They are explained below in detail:

Standard Forms

The standard form in Boolean algebra is expressed in terms of Sum of Products (SOP) and Product of Sums (POS). These are the two types of standard forms.

• Sum of Products (SOP)

This is an expression in which each term is a product term and all the product terms are summed together. It is a Boolean expression containing AND terms, called product terms, of one or more literals each. This is an OR operation.

i. Minimal SOP

An expression in which each product term consists of the minimum number of variables is minimal SOP.
For example; XY+XY̅+X̅Y

ii. Maximal SOP

It is the expression in which each product term consists of the maximum number of variables.
For example; XYZ+X̅YZ+XY̅Z

• Product of Sums (POS)

This is an expression in which each sum term several sum terms multiplied together. It is a Boolean expression that contains OR terms, called sum terms. It is an AND operation. Each term may have many numbers of literals.

i. Minimal POS

An expression in which each of the sum terms consists of a minimum number of variables is denoted as minimal POS.
For example; (X+Y) (X̅+Y) (X+Y̅)

ii. Maximal POS

This is an expression in which each sum term contains a maximum number of variables.
For example: (W+X+Y+Z) (W̅+X̅+Y̅+Z) (W+X̅+Y̅+Z)

Canonical Forms

The form in which the variables are represented in terms of max terms and minterms is called a canonical form. It is useful in analysis and design.

• Minterms

When a sum of product form of logic expression is in canonical form, each product term is called minterm. Each product term contains all the variables.
The minterm is denoted by ‘m’.

• Max terms

When a product of the sum term is in canonical form, each term is called the max term. It also contains all the variables.
It is denoted by ‘M’.

Minterms and Maxterms of three variables:

X	Y	Z	Minterms	Notation	Maxterms	Notation
0	0	0	X̅ Y̅ Z̅	m0	X+Y+Z	M0
0	0	1	X̅ Y̅ Z	m1	X+Y+Z̅	M1
0	1	0	X̅ Y Z̅	m2	X+Y̅+Z	M2
0	1	1	X̅ Y Z	m3	X+Y̅+Z̅	M3
1	0	0	X Y̅ Z̅	m4	X̅+Y+Z	M4
1	0	1	X Y̅ Z	m5	X̅+Y+Z̅	M5
1	1	0	X Y Z̅	m6	X̅+Y̅+Z	M6
1	1	1	X Y Z	m7	X̅+Y̅+Z̅	M7