# Sum of Products and Product of Sums Expressions

~ on ~ 0 commentsFigure 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)

*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*= (

*A*

**∙**

*B*) + (

*C*

**∙**

*D*),

**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.

*product-of-sums*(POS) expression, which can be written as

*V*= (

*A*+

*B*)(

*C*+

*D*)

*(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.

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