notes/education/computer engineering/ECE2700/Karnaugh Maps.md

A Karnaugh map is an alternative to a truth table for representing a function in boolean algebra, and serve as a way to derive minimum cost circuits for a truth table.

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Given the above truth table, the columns are labelled with x_1, and the rows are labelled with x_2.

To find a minimal boolean expression with a Karnaugh map, we need to find the smallest number of product terms (x_1, x_2) that should produce a 1 for all instances where the cell in a table is 1.

Two Variable Maps

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• Given the map described in the above image, the output is 1 for the row where x_2 is equal to 1.
• Similarly, the output is 1 for the column where x_1 is equal to zero.
• By ORing the condition where x_1 is zero (\overline{x_1}), and the condition where x_2 is one (x_1), we can find a minimal expression for the truth table.

Three Variable Maps

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A three variable Karnaugh map is constructed by placing 2 two-variable maps side by side. The values of x_1 and x_2 distinguish columns in the map, and the value of x_3 distinguishes rows in the map.

To convert a 3 variable Karnaugh map to a minimal boolean expression, start by looking for places in the map that contain 1s next to each other (by row, or by column).

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From there, describe the pair of 1s using boolean a