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A Boolean function specified by a truth table can be expressed algebraically in many different ways. By manipulating a Boolean expression according to Boolean algebra rules, one may obtain a simpler expression that will require fewer gates. To see how this is done, we must first study the manipulative capabilities of Boolean algebra.

Table 1-1 lists the most basic identities of Boolean algebra. All the identities in the table can be proven by means of truth tables. The first eight identities show the basic relationship between a single variable and itself, or in conjunction with the binary constants 1 and 0. The next five identities (9 through 13) are similar to ordinary algebra. Identity 14 does not apply in ordinary algebra but is very useful in manipulating Boolean expressions. Identities 15 and 16 are called DeMorgan's theorems and are discussed below. The last identity states that if a variable is complemented twice, one obtains the original value of the variable.

Table 1-1

The identities listed in the table apply to single variables or to Boolean functions expressed in terms of binary variables. For example, consider the following Boolean algebra expression:

AB' + C'D + AB' + C'D

By letting x = AB' + C'D the expression can be written as x + x. From identity 5 in Table 1-1 we find that x + x = x. Thus the expression can be reduced to only two terms:

AB' + C' D+ A' B+ C' D= AB' + C' D

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