Boolean Equations

In this section, our purpose is to write a Boolean equation for any logic function given its truth table.

Terminology

1

Minterm and Maxterm

• A minterm is a product involving all of the inputs to the function.

• A maxterm is a sum involving all of the inputs to the function.

e.g., $A\bar B\bar C$ is a minterm, $A+\bar B+C$ is a maxterm for a function of the three variables A, B, C.

There is one very important property regarding minterm and maxterm. It states that

There exists one and only one input value such that given minterm = 1 / maxterm = 0.

This property ensures that when writing the boolean equation from a truth table, we can only do the followings:

1. For Sum of Product: We sum each of the minterms of which the output is TRUE.

2. For Product of Sum: We multiply each of the maxterms of which the output is FALSE.

2

The order of oeprations

The order of operations is important when interpreting Boolean equations. In Boolean equations, NOT has the highest precedence, followed by AND, then OR.

Sum-of-Products Form

We can write a Boolean equation for any truth table by summing each of the minterms for which the output is TRUE. And this boolean equation is called the sum-of-products canonical form of a function because it is the sum (OR) of products (ANDs forming minterms)

Product-of-Sums Form

An alternative way of expressing Boolean functions is the product-of-sums canonical form. Each row of a truth table corresponds to a maxterm that is FALSE for that row.

SoP and PoS, which one to choose?

It depends on there are how many TRUEs and FALSEs in your truth table.

Sum-of-products (SoP) produces the shortest equations when the output is TRUE on only a few rows of a truth table; product-of-sums (PoS) is simpler when the output is FALSE on only a few rows of a truth table.