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# Sum of Products and Product of Sums Expressions

Figure 2.14 shows an AND-OR gate using four mechanical switches. First we connect switches A and B in series. Similarly, switches C and D are also connected in series, as shown. These series combinations are then connected in parallel to each other. A resistive load R is connected to this parallel combination. Current I will flow through the load, only when one pair of the switches (i.e., either the A and B combination or the C and D combination), is closed simultaneously; then the output voltage Z = IR = +V volts. The same result is obtained when all the four switches are closed simultaneously. However, when one (or both) of the switches in both the pairs are open, no current will flow through the load, and hence the output voltage Z = 0 volt. This represents an AND-OR operation. We may write the Boolean expression in this case as

V= (AB) + (CD)    (2.6)

Equation (2.6) is to be read as V = (A AND B) OR (C AND D). This means that load current will flow whenever A and B or C and D are closed simultaneously. This Boolean expression, which states that Z = (AB) + (CD),  is called a sum-of-products  (SOP) expression, as it represents the sum of two Boolean algebraic product terms. It is also called the canonical form of SOP expression.

In a similar fashion, using complementary terms, we can form the product-of-sums (POS) expression, which can be written as

V = (A + B)(C + D)    (2.7)

The switching circuit shown (example) in Fig. 2.15 represents the expression given in Eq. (2.7).

A logic expression may be obtained from the truth table by combining the terms for which the output is a logic 1.  This results in an SOP expression, which can be seen to be the sum (OR) of several product (AND) functions.  As an example, consider the logic expression

Z = ABC+A′BC′         (2.8)

Here, ABC and A′BC′ are two logical AND functions; ORing of these products produces output Z.