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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
1for the row wherex_2is equal to 1. - Similarly, the output is
1for the column wherex_1is equal to zero. - By ORing the condition where
x_1is zero (\overline{x_1}), and the condition wherex_2is 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 algebra.
In the above example, the top pair of 1s is in the column where x_3 is equal to zero (\overline{x_3}), and x_1 is equal to 1 (x_1). This describes a single term in the resulting equation (x_1\overline{x_3}).
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Similar logic can be employed using more than just a pair of ones.
Four Variable Maps
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