8 Understanding Minterms and Maxterms

Aug 28, 2024

Lecture 8: Minterms and Maxterms

Key Topics Covered

  • Review of Minterms
  • Introduction to Maxterms
  • Differences between Minterms and Maxterms
  • Use of De Morgan's Theorems
  • Truth Tables

Minterms

  • Definition: A minterm is a specific type of product of all Boolean variables, where each variable is either in its complemented or uncomplemented form.
  • Expression: The minterm expansion is expressed as a sum of products.
  • Notation: Minterms are denoted using lowercase m, e.g., m2, m5, m7.
  • Identification: Minterms are identified based on when they are equal to 1.
    • Example: For function f(X, Y, Z), m2 is when X=0, Y=1, Z=0, etc.
  • Concept: Any minterm expansion is a sum of products (SOP) expression.
  • Special Case: If a term does not include all three variables, it is not a minterm.

Maxterms

  • Definition: A maxterm is a specific type of sum of all Boolean variables, where each variable is either in its complemented or uncomplemented form.
  • Expression: The maxterm expansion is expressed as a product of sums.
  • Notation: Maxterms are denoted using uppercase M, e.g., M3, M4, M5.
  • Identification: Maxterms are identified based on when they are equal to 0.
    • Example: M3 is when X=0, Y=1, Z=1, etc.
  • Concept: Any maxterm expansion is a product of sums (POS) expression.

Relationship Between Minterms and Maxterms

  • De Morgan’s Theorems:
    • The complement of a minterm (e.g., m6) is the corresponding maxterm (e.g., M6).
    • The complement of a product is the sum of individual complements and vice versa.
  • Example:
    • m6' = M6
    • M6' = m6

Truth Tables

  • Purpose: Used to simplify the identification of minterms and maxterms.
  • Construction: List all possible combinations of variables and indicate when the function is 0 or 1.
  • Example: For G(X, Y, Z), identified by minterm and maxterm based on when the function is 1 or 0 respectively.

Converting Between Minterms and Maxterms

  • Method:
    • Use De Morgan's laws to convert between SOP (sum of products) and POS (product of sums).
    • Can be computed directly from the truth table without algebra.
  • Example Problem: Given a function G, find the minterm and maxterm expansions using the truth table.
  • Verification: Algebraic verification using De Morgan's theorems can be used to confirm calculations.

Problem-Solving Strategy

  1. Identify the terms using truth tables.
  2. Apply De Morgan’s Theorems for proof and conversion.
  3. Verify by ensuring no repeated terms in expansions.

Test Questions

  • 8.1: Find minterm expansion for given function.
  • 8.2: Find maxterm expansion for the same function.

This lecture provided a comprehensive review of minterms and introduced maxterms, highlighting their differences and how to utilize them through truth tables, with practical examples and test questions for practice.