How to use a dual to find the complement of a boolean expression

Duality refers to the relationship between two mathematical objects that are, in some sense, opposite to each other. A dual mathematical expression is obtained by reversing certain operations and properties.

For example, the dual of the following mathematical expression can be found by changing the multiplication operator with the addition operator and vice versa.

Original expression:

Dual of the expression:

Complement of a boolean expression

The complement represents a negated version of the original expression. It is commonly denoted by placing a bar on top of the expression (which actually represents the $NOT$ of the expression).

The truth table of the expression $f = A.\overline{B}$ and its complement $\overline{f} = \overline{A.\overline{B}}$ is as follows:

Duality principles

We can use the following principles to find the dual of a boolean expression which will eventually give us its complement.

• Change all the $OR$ operations to $AND$ and vice versa.

• Convert all the true ($1$) values to false ($0$) and vice versa.

• Apply the $NOT$ operation on every literalEach term used in a boolean expression is called a literal..

By applying all the above duality principles on $f$, we can obtain the expression for $\overline{f}$:

Changing the $OR$ operation to $AND$ :

Applying the $NOT$ operation on every literal:

We can examine the outputs of the resultant expression using the following truth table:

From the last two columns, we can verify that the expression we got by taking the dual of the boolean expression $f$ is its complement.

Advantage

Finding the complement of a small Boolean expression may be relatively easier. But as the expression becomes longer and more complex, determining the complement using the truth tables or Boolean theorems becomes tedious.

In such cases, the principles of duality provide a more convenient approach to finding the complement of an expression. Duality provides a shortcut to the complement without going through extensive mathematical calculations.

Let's take the dual of the following expression to find its complement:

Changing the $OR$ operations to $AND$ and vice versa:

Applying the $NOT$ operation to each literal:

Converting the rightmost true (1) value to false (0):

The above expression is the complement of $y$. You can verify it by making its truth table. The resultant expression can further be simplified using the idempotence theorem, DeMorgan's law, and other identities.