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4.2 Singles, pairs, quads, octs, etc

A “single” is a cell that has a value of 1. In our example, we have five of these.

Next, we try to identify groups of two singles (vertical or horizontal). Note that variable X wraps around from the last column to the first column. We can identify four pairs. In table 5, we represent the membership of a cell to each pair by a set. The pairs are identified as a,b,c,d.

Z ↓ XY →00 01 11 10
0 1 {d} 1 {a,c}1 {a,d}
1 1 {b}1 {b,c}

Table 5: K-map for table 1, with cell group membership identified.

Note that each adjacent pair represents that a variable is not significant in the product term. We have, thus, identified four methods to reduce the equation.

The next step is to identify pairs of pairs. In other words, we want to locate two pairs that are adjacent to each other. In this table, however, we do not have such cases. If the cell corresponding to XY = 01,Z = 0 is also a 1, then the entire first row (consisting of two pairs) can be combined.

Because we cannot create any pairs of pairs, then there is no need to look for pairs of pairs of pairs.

Once we can no longer combine pairs into larger units, we need to select the smallest number of singles, pairs, pairs of pairs and etc.

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