Karnaugh Maps

Discrete Math · Axiom Academy

A visual method for systematically simplifying Boolean expressions by identifying and grouping adjacent minterms in a grid structure. A Karnaugh Map is a 2D grid where each cell represents a minterm (a unique combination of input variables). The key property: adjacent cells differ by exactly one variable . Each cell corresponds to one row in a truth table Cells are arranged so adjacent cells have one bit change (Gray code ordering) Cells contain the output value (0 or 1) for that input combination The grid "wraps around" - edges are considered adjacent The simplest K-map has 2 variables (A, B) creating a 2×2 grid with 4 cells. Let's see how to use it to simplify a Boolean function. Example: Consider the function where the output is 1 when A=0,B=1 or A=1,B=1. The animation shows how grouping these adjacent 1s eliminates variable A, giving us the simplified expression: F = B With 3 variables (A, B, C), we get a 2×4 grid with 8 cells. Notice the column ordering: 00, 01, 11, 10 - this Gray code ensures adjacent columns differ by one bit. The animation demonstrates finding a prime implicant - a maximal group of 1s that cannot be made larger. Each group gives us one term in the simplified expression. With 4 variables (A, B, C, D), we use a 4×4 grid with 16 cells. Both rows and columns use Gray code ordering, and the map wraps in both directions.

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