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Minterms of adjacent squares in the map are identical except for one variable, which appears complemented in one square and uncomplemented in the adjacent square. According to this definition of adjacency, the squares at the extreme ends of the same horizontal row are also to be considered adjacent. The same applies to the top and bottom squares of a column. As a result, the four corner squares of a map must also be considered to be adjacent.

A Boolean function represented by a truth table is plotted into the map by inserting 1's in those squares where the function is 1. The squares containing 1's are combined in groups of adjacent squares. These groups must contain a number of squares that is an integral power of 2. Groups of combined adjacent squares may share one or more squares with one or more groups. Each group of squares represents an algebraic term, and the OR of those terms gives the simplified algebraic expression for the function. The following examples show the use of the map for simplifying Boolean functions.

In the first example we will simplify the Boolean function

F(A, B, C) =?(3, 4, 6, 7)

The three-variable map for this function is shown in Fig. 1-8. There are four squares marked with 1's, one for each minterm that produces 1 for the function. These squares belong to minterms 3, 4, 6, and 7 and are recognized from Fig. 1-7(b). Two adjacent squares are combined in the third column. This column belongs to both B and C and produces the term BC. The remaining two squares with 1's in the two corners of the second row are adjacent and belong to row A and the two columns of C', so they produce the term AC'. The simplified algebraic expression for the function is the OR of the two terms:

F = BC + AC'

The second example simplifies the following Boolean function: F(A, B, C)= ? (0, 2, 4, 5, 6)

The five minterms are marked with 1's in the corresponding squares of the three-variable map shown in Fig. 1-9. The four squares in the first and fourth columns are adjacent and represent the term C'. The remaining square marked with a 1 belongs to minterm 5 and can be combined with the square of minterm 4 to produce the term AB'. The simplified function is

F=C'+AB'

The third example needs a four-variable map.

F(A, B, C, D) =? (0, 1, 2, 6, 8, 9,10)

The area in the map covered by this four-variable function consists of the squares marked with 1's in Fig. 1-10. The function contains 1's in the four corners that, when taken as a group, give the term B'D'. This is possible because these four squares are adjacent when the map is considered with top and bottom or left and right edges touching. The two 1's on the left of the top row are combined with the two 1's on the left of the bottom row to give the term B'C'. The remaining 1 in the square of minterm 6 is combined with minterm 2 to give the term A'CD'. The simplified function is

F = B'D' + B'C' + A'CD'

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