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Each combination of the variables in a truth table is called a minterm. For example, the truth table of Fig. 1-3 contains eight minterms. When expressed in a truth table a function of n variables will have 2" minterms, equivalent to the 2" binary numbers obtained from n bits. A Boolean function is equal to 1 for some minterms and to 0 for others. The information contained in a truth table may be expressed in compact form by listing the decimal equivalent of those minterms that produce a 1 for the function. For example, the truth table of Fig. 1-3 can be expressed as follows:

Figure 1-3

F(x,y,z) =? (1,4,5,6,7)

The letters in parentheses list the binary variables in the order that they appear in the truth table. The symbol 1~ stands for the sum of the minterms that follow in parentheses. The minterms that produce 1 for the function are listed in their decimal equivalent. The minterms missing from the list are the ones that produce 0 for the function. The map is a diagram made up of squares, with each square representing one minterm. The squares corresponding to minterms that produce 1 for the function are marked by a 1 and the others are marked by a 0 or are left empty. By recognizing various patterns and combining squares marked by 1's in the map, it is possible to derive alternative algebraic expressions for the function, from which the most convenient may be selected.

Figure 1-7

The maps for functions of two, three, and four variables are shown in Fig. 1-7. The number of squares in a map of n variables is 2". The 2" minterms are listed by an equivalent decimal number for easy reference. The minterm numbers are assigned in an orderly arrangement such that adjacent squares represent minterms that differ by only one variable. The variable names are listed across both sides of the diagonal line in the corner of the map. The 0's and 1's marked along each row and each column designate the value of the variables. Each variable under brackets contains half of the squares in the map where that variable appears unprimed. The variable appears with a prime (complemented) in the remaining half of the squares.

The minterm represented by a square is determined from the binary assignments of the variables along the left and top edges in the map. For example, minterm 5 in the three-variable map is 101 in binary, which may be obtained from the 1 in the second row concatenated with the 01 of the second column. This minterm represents a value for the binary variables A, B, and C, with A and C being unprimed and B being primed (i.e., AB'C). On the other hand, minterm 5 in the four-variable map represents a minterm for four variables. The binary number contains the four bits 0101, and the corresponding term it represents is A'BC'D.