# Forms and Definitions of Boolean Expressions

Numerical Representation

### Numerical Representation

Take as an example the truth table of a three-variable function as shown below. Three variables, each of which can take the values 0 or 1, yields eight possible combinations of values for which the function may be true. These eight combinations are listed in ascending binary order and the equivalent decimal value is also shown in the table.

 DecimalValue A B C f 0 0 0 0 1 1 0 0 1 0 2 0 1 0 1 3 0 1 1 1 4 1 0 0 0 5 1 0 1 0 6 1 1 0 0 7 1 1 1 1

The function has a value 1 for the combinations shown, therefore:

......(1)

This can also be written as:
f(A, B, C) = 000 + 010 + 011 + 111

Note that the summation sign indicates that the terms are "OR'ed" together. The function can be further reduced to the form:

f(A, B, C) = (000, 010, 011, 111)

It is self-evident that the binary form of a function can be written directly from the truth table.
Note:

(a) the position of the digits must not be changed
(b) the expression must be in standard sum of products form.

It follows from the last expression that the binary form can be replaced by the equivalent decimal form, namely:

f(A, B, C) = (0,2,3,7)......(2)

### Product of Sums Representation

From the truth table given above the function has the value 0 for the combinations shown, therefore

......(3)

Writing the inverse of this function:

Applying De Morgan's Theorem we obtain:

Applying the second De Morgan's Theorem we obtain:

......(4)

The function is expressed in standard product of sums form.

Thus there are two forms of a function, one is a sum of products form (either standard or normal) as given by expression (1), the other a product of sums form (either standard or normal) as given by expression (4). The gate implementation of the two forms is not the same!

### Examples

Consider the function:

In binary form: f(A, B, C, D) = (0101, 1011, 1100, 0000, 1010, 0111)
In decimal form: f(A, B, C, D) = (5, 11, 12, 0, 10, 7)

### Problem

Compare expressions (3) and (4): what can you deduce that will enable you in future to write down directly the product of sums form given the inverse of the function.