📜 Lecture 4: Minterms and Maxterms

This lecture explains how to convert a truth table into a Boolean equation using two canonical forms: Sum of Products (SOP) and Product of Sums (POS).

Key Concepts

• Goal: To convert a truth table into a logical expression, which can then be built as a circuit.

Minterms (m) and Sum of Products (SOP)

• Minterm: A product (AND) term that corresponds to a single row of the truth table where the function output F=1.
• Rule: To write a minterm, if an input variable is 0, it is complemented (e.g., A'). If it is 1, it is not complemented (e.g., A).
• Example:
  • Input A=0, B=1, C=0 (Decimal 2) -> Minterm m2 = A'BC'.
  • Input A=1, B=0, C=1 (Decimal 5) -> Minterm m5 = AB'C.
• Sum of Products (SOP): The final function F is the sum (OR) of all the minterms for which F=1.
• Example (Prime Detector):
  • From Lecture 3, F=1 for minterms m(2, 3, 5, 7).
  • SOP Equation: F = m2 + m3 + m5 + m7
  • F = (A'BC') + (A'BC) + (AB'C) + (ABC)
• Definitions:
  • Canonical SOP: The full SOP expression with all minterms.
  • On-Set: The list of minterms where F=1. e.g., F(A,B,C) = Σm(2, 3, 5, 7).
• Circuit: An SOP expression is implemented using an AND-OR circuit (a layer of AND gates feeding into a single OR gate).

Maxterms (M) and Product of Sums (POS)

• Maxterm: A sum (OR) term that corresponds to a single row of the truth table where the function output F=0.
• Rule: (Opposite of minterms) If an input variable is 0, it is not complemented (e.g., A). If it is 1, it is complemented (e.g., A').
• Example:
  • Input A=0, B=0, C=1 (Decimal 1) -> Maxterm M1 = A + B + C'.
  • Input A=1, B=1, C=0 (Decimal 6) -> Maxterm M6 = A' + B' + C.
• Product of Sums (POS): The final function F is the product (AND) of all the maxterms for which F=0.
• Example (Prime Detector):
  • From Lecture 3, F=0 for maxterms M(0, 1, 4, 6).
  • POS Equation: F = M0 · M1 · M4 · M6
  • F = (A+B+C) · (A+B+C') · (A'+B+C) · (A'+B'+C)
• Definitions:
  • Canonical POS: The full POS expression with all maxterms.
  • Off-Set: The list of maxterms where F=0. e.g., F(A,B,C) = ΠM(0, 1, 4, 6).
• Circuit: A POS expression is implemented using an OR-AND circuit (a layer of OR gates feeding into a single AND gate).

Circuit Implementation Notes

• SOP (AND-OR) circuits can be converted to NAND-NAND circuits.
• POS (OR-AND) circuits can be converted to NOR-NOR circuits.
• NAND and NOR gates are often cheaper, faster, and smaller than AND/OR gates.