Representation of Function in Sum of Minterms or Product of Maxterms

Representation of Boolean Function in Sum of Minterms or Product of Maxterms

In digital logic, a Boolean function can be expressed in two canonical forms:

1. Sum of Minterms (SOM): Also known as the Disjunctive Normal Form (DNF).
2. Product of Maxterms (POM): Also known as the Conjunctive Normal Form (CNF).

Both forms provide a systematic way to express any Boolean function, and both are particularly useful for simplifying logic circuits.

1. Sum of Minterms (SOM)

The Sum of Minterms is a Boolean expression that represents the OR of all possible minterms (ANDed terms), where each minterm corresponds to a combination of the variables that makes the function true (i.e., the function evaluates to 1). The sum of minterms is the OR of these minterms.

Minterm:

• A minterm is a product (AND operation) of all the variables in the Boolean function, where each variable appears either in its original form or in its complement (NOT form).
• A minterm corresponds to a single row in the truth table where the output is 1.

Example:

Consider the Boolean function $F(A, B, C)$ defined as:

The minterms where $F(A, B, C) = 1$ are:

• Row 1: $A = 0, B = 0, C = 0$ → Minterm = $\overline{A} \, \overline{B} \, \overline{C}$
• Row 3: $A = 0, B = 1, C = 0$ → Minterm = $\overline{A} \, B \, \overline{C}$
• Row 4: $A = 0, B = 1, C = 1$ → Minterm = $\overline{A} \, B \, C$
• Row 7: $A = 1, B = 1, C = 0$ → Minterm = $A \, B \, \overline{C}$

Thus, the Sum of Minterms (SOM) for this function is:

This is the canonical Sum of Minterms representation.

2. Product of Maxterms (POM)

The Product of Maxterms is a Boolean expression that represents the AND of all possible maxterms, where each maxterm corresponds to a combination of variables that makes the function false (i.e., the function evaluates to 0). The product of maxterms is the AND of these maxterms.

Maxterm:

• A maxterm is a sum (OR operation) of all the variables in the Boolean function, where each variable appears either in its original form or in its complement (NOT form).
• A maxterm corresponds to a single row in the truth table where the output is 0.

Example:

Using the same Boolean function $F(A, B, C)$ defined above:

The maxterms where $F(A, B, C) = 0$ are:

• Row 2: $A = 0, B = 0, C = 1$ → Maxterm = $A \, \lor \, B \, \lor \, \overline{C}$
• Row 5: $A = 1, B = 0, C = 0$ → Maxterm = $\overline{A} \, \lor \, B \, \lor \, C$
• Row 6: $A = 1, B = 0, C = 1$ → Maxterm = $\overline{A} \, \lor \, B \, \lor \, \overline{C}$
• Row 8: $A = 1, B = 1, C = 1$ → Maxterm = $\overline{A} \, \lor \, \overline{B} \, \lor \, \overline{C}$

Thus, the Product of Maxterms (POM) for this function is:

This is the canonical Product of Maxterms representation.

Summary of the Canonical Forms:

• Sum of Minterms (SOM):

  • The function is expressed as the OR of all minterms (ANDed terms) where the output is 1 in the truth table.
  • The minterm for each row is formed by ANDing all variables, using the variable or its complement based on the output.

• Product of Maxterms (POM):

  • The function is expressed as the AND of all maxterms (ORed terms) where the output is 0 in the truth table.
  • The maxterm for each row is formed by ORing all variables, using the variable or its complement based on the output.

Why Use Canonical Forms?

1. Simplicity in Design: Canonical forms are useful because they provide a direct method for implementing Boolean functions in digital circuits, especially when dealing with truth tables.
2. Simplification: These forms are often the starting point for simplifying complex Boolean expressions using techniques like Boolean algebra or Karnaugh maps.
3. Automation: Automated tools and software use canonical forms to design logic circuits based on truth tables, making it easier to implement digital systems.