Minterm Expansion: A unique way of expressing Boolean functions as a sum of products of all available variables in complemented or uncomplemented form.
Decimal Notation: Represents the combination of Boolean variables in decimal form instead of algebraic notation.
Main Ideas
Minterm Expansion of f(x, y, z)
Example:
f(x, y, z) = x'y'z + xyz' is a minterm expansion.
Minterm expansions are derived based on when they equal 1.
Algebraic vs Decimal Notation:
Algebraic Notation: Variables spelled out.
Decimal Notation: Each binary number interpreted as a decimal (e.g., 011 is 3).
Importance of Variable Order
Ensure variables in minterms match the order in the function’s argument.
Misordering can lead to incorrect minterm/maxterm conversions, as shown with f(x, y, z).
Problem Example
Given: f(A, B, C, D) = AB + A'CD
Goal: Find the minterm and maxterm expansions.
Finding Minterm Expansion
Use algebraic manipulations to make each term a minterm:
Example: AB becomes ABCD using (C + C')(D + D').
Result: Expand into minterms ABCD, ABC'D', etc.
Finding Maxterm Expansion
Approach 1: Use algebraic manipulations, convert SOP to POS using De Morgan’s Theorems.
Approach 2: Utilize minterm expansion to identify maxterms by determining where the function is 0 based on truth table implications.
Alternative Method
A more efficient method to find maxterm expansion:
Derive minterms first, convert to decimal notation.
Identify missing numbers in the decimal notation for maxterms.
Exercise
Practice problems from Roth’s "Fundamentals of Logic Design", discussing finding both expansions.
Use decimal notation for both minterm and maxterm expansions.
Assignments
Problem 9.1: Given minterms, identify which is the minterm expansion for f'.
Problem 9.2: Find the maxterm expansion for a given formula in decimal notation.
Conclusion
Understanding and applying these concepts ensures accurate function representation in both algebraic and decimal forms. Make use of alternative methods for efficiency and accuracy in logic design.