Canonical and Standard Forms

All Boolean expressions, regardless of their form, can be converted into either of two standard forms:

  • Sum of the Products (SOP)
  • Product of the Sums (POS)
Standardization makes the evaluation, simplification, and implementation of Boolean expressions more systematic and easier.

A Boolean function can be uniquely described by its truth table, or in one of the canonical forms. A canonical form specifies a unique representation of the Boolean function, while a normal form simply specifies its form, without the requirement of uniqueness.

Canonical Sum of Products Form

There are 2 steps to derive the Canonical Sum of Products Form from its truth table.

  1. A Minterm is a product (AND) term containing all input variables of the function in either true or complemented form. A variable appears in complemented form ~X if it is a 0 in the row of the truth-table, and as a true form X if it appears as a 1 in the row.
    • Examples
    • A=0, B=1, C=0 -> ~A*B*~C
    • A=1. B=0, C=0 -> A*~B*~C
  2. The canonical form is obtained by taking the sum (OR) of the minterm of the rows where a 1 appears in the output.
  1. This truth table is interactive.
  2. Click on the row output to change its value and observe the change in the Boolean Expression.
  3. The minterm is included in the Boolean Expression only if the output of that row is 1

The Boolean expression of any combinational logic circuit described by a truth-table can be obtained by writing its canonical form. Use Boolean Algebra or Karnaugh Map to simplify the expression which can then be translated to a circuit of logic gates.

Learn by Doing

Design an Odd Number Detector to deepen your understanding of the Sum of Products.

An Odd Number Detector outputs a 1 whenever there is an odd value in its inputs.