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DeMorgan's theorem is very important in dealing with NOR and NAND gates. It states that a NOR gate that performs the (x + y)' function is equivalent to the function x'y'. Similarly, a NAND function can be expressed by either (xy)' or (x' + y'). For this reason the NOR and NAND gates have two distinct graphic symbols, as shown in Figs. 1-4 and 1-5. Instead of representing a NOR gate with an OR graphic symbol followed by a circle, we can represent it by an AND graphic symbol preceded by circles in all inputs. The invert-AND symbol for the NOR gate follows from DeMorgan's theorem and from the convention that small circles denote complementation. Similarly, the NAND gate has two distinct symbols, as shown in Fig. 1-5.

Figure 1-4 , 1-5

To see how Boolean algebra manipulation is used to simplify digital circuits, consider the logic diagram of Fig. 1-6(a). The output of the circuit can be expressed algebraically as follows:

F = ABC + ABC' + A'C

Each term corresponds to one AND gate, and the OR gate forms the logical sum of the three terms. Two inverters are needed to complement A' and C'. The expression can be simplified using Boolean algebra.

F = ABC + ABC' + AC = AB(C + C') + A'C = AB + A'C

Note that (C + C)' = 1 by identity 7 and AB • 1= AB by identity 4 in Table 1-1.

The logic diagram of the simplified expression is drawn in Fig. 1-6(b). It requires only four gates rather than the six gates used in the circuit of Fig. 1-6(a). The two circuits are equivalent and produce the same truth table relationship between inputs A, B, C and output F.