Canonical and Standard Forms

Part 1: Introduction

In Boolean algebra, canonical and standard forms are specific ways to represent Boolean expressions. They provide a structured and consistent method for analyzing and simplifying logical functions.

Part 2: Conversion of truth table to Boolean Expression

A Boolean function is a mathematical expression that operates on binary variables, producing a binary output. Truth tables provide a tabular representation of a Boolean function, listing all possible input combinations and their corresponding outputs. To algebraically represent a Boolean function from a truth table, one can employ Sum-of-Products (SOP) or Product-of-Sums (POS) expressions. In SOP, product terms are created for input combinations that yield a 1 output, and these terms are then summed. Conversely, in POS, sum terms are formed for input combinations that result in a 0 output, and these terms are multiplied. Both methods produce equivalent representations of the same Boolean function.

Part 2.1 SUM-OF-PRODUCTS and PRODUCT-OF-SUMS Expressions

To effectively derive Boolean expressions, a strong foundation in fundamental concepts is crucial. These concepts include product terms, which represent the AND operation of variables, and sum terms, which represent the OR operation of variables. Sum-of-Products (SOP) and Product-of-Sums (POS) expressions are common forms of Boolean expressions, composed of sums or products of these terms, respectively. Understanding these concepts empowers you to make informed decisions about the most suitable gate networks for implementation, leading to efficient and effective digital circuit design.

Part 2.1.1 Two Types of Terms

Part 2.1.1.1 Product Term or Minterm

In Boolean algebra, a product term is a logical expression that represents the AND operation of one or more variables. It consists of variables, possibly complemented (inverted), connected by AND operators.

Part 2.1.1.2 Sum Term or Maxterm

In Boolean algebra, a sum term is a logical expression that represents the OR operation of one or more variables. It consists of variables, possibly complemented (inverted), connected by OR operators.

Input			Product Term or Minterm	Sum Term or Maxterm
A	B	C
0	0	0	A’B’C’	A + B + C
0	0	1	A’B’C	A + B + C’
0	1	0	A’BC’	A + B’ + C
0	1	1	A’BC	A + B’ + C’
1	0	0	AB’C’	A’ + B + C
1	0	1	AB’C	A’ + B + C’
1	1	0	ABC’	A’ + B’ + C
1	1	1	ABC	A’ + B’ + C’

Part 2.1.2 Two Types of Expression

Part 2.1.2.1 Sum-Of-Product Expression

A Sum-of-Products (SOP) expression is a Boolean expression that represents a logical function as the sum (OR) of product terms. Each product term represents a combination of input variables that produces a 1 (true) output.

Part 2.1.2.2 Product-Of-Sum Expression

A Product-of-Sums (POS) expression is a Boolean expression that represents a logical function as the product (AND) of sum terms. Each sum term represents a combination of input variables that produces a 0 (false) output.

Example No. 1

Convert the truth table below to Boolean Expression

Input			Output
A	B	C	F
0	0	0	0
0	0	1	0
0	1	0	1
0	1	1	0
1	0	0	1
1	0	1	0
1	1	0	1
1	1	1	1

Solution:

Add first the table of minterm and maxterm for easy determination.

Input			Output	Product Term or Minterm	Sum Term or Maxterm
A	B
0	0	0	0	A’B’C’	A + B + C	0
0	0	1	0	A’B’C	A + B + C’	1
0	1	0	1	A’BC’	A + B’ + C	2
0	1	1	0	A’BC	A + B’ + C’	3
1	0	0	1	AB’C’	A’ + B + C	4
1	0	1	0	AB’C	A’ + B + C’	5
1	1	0	1	ABC’	A’ + B’ + C	6
1	1	1	1	ABC	A’ + B’ + C’	7

Next, we need to decide what we will use, SOP or POS base on the number of 0’s and 1’s. But for the sake of this discussion we will derive both SOP and POS.

Derive SOP.

Collect the number of 1’s output, then create the formula.

F = A’BC’ + AB’C’ +ABC’ + ABC

This is our answer.

Derive POS.

Collect the number of 0’s output, then create the formula.

F = (A + B + C)(A + B + C’)(A + B’ +C’)(A’ + B + C’)

This is our answer.

Other representation of SOP is using the symbol 

F (A, B, C) = (2, 4, 6, 7)

Other representation of POS is using the symbol 

F (A, B, C) = (0, 1, 3, 5)