Product of Summation and Summation of Product

Product of Summation and Summation of Product

SOP and POS

The terms “product” and “sum” have been borrowed from mathematics to describe AND and OR logic operations. Any logic system can be represented in one of these two logic ways. As it will be explained in this section, the two forms are equivalent ways of expressing a logic system; however, some logic systems lend themselves to one rather than the other.

A product term is defined as an AND relationship between any number of variables, and a sum term is defined as an OR relationship between any number of logic variables. Any logic system can be represented in two logically equivalent ways: as the OR'ing of AND'ed terms, known as the Sum of Products (SOP) form; or as the AND'ing of OR'ed terms, known as the Product of Sums (POS) form. The two forms are interchangeable, and one form can be transformed to the other following a few basic rules. As an example, consider the XOR relationship YSOP: \[Y_{SOP}=\overline{A}\cdot B + A \cdot \overline{B}\] This SOP relationship can be expressed in POS form as: \[Y_{POS}=(A+B)\cdot (\overline{A}+\overline{B})\] In this example, the POS and SOP forms are equally simple, but this is not always the case. For circuits with more than two inputs, it may turn out that one form is simpler that the other. If a circuit is to be constructed, it makes sense to evaluate both forms so that the simplest one can be constructed.

Construct SOP and POS From a Truth Table

A logic equation (and therefore a logic circuit) can easily be constructed from any truth table by applying the rules presented below.

For SOP circuits:

  1. A circuit for a truth table with N input columns can use AND gates with N inputs, and each row in the truth table with a '1' in the output column requires one N-input AND gate.
  2. Inputs to the AND gate are inverted if the input shows a '0' on the row, and not inverted if the input shows a '1' on the row.
  3. All AND terms are connected to an M-input OR gate, where M is the number of '1' output rows.
  4. The output of the OR gate is the function output.
Figure 1. Construct SOP circuit from a truth table.

For POS circuits:

  1. A circuit for a truth table with N input columns can use OR gates with N inputs, and each row in the truth table with a '0' in the output column requires one N-input OR gate.
  2. Inputs to the OR gate are inverted if the input shows a '1' on the row, and not inverted if the input shows a '0' on the row.
  3. All OR terms are connected to an M-input AND gate, where M is the number of '1' output rows.
  4. The output of the AND gate is the function output.
Figure 2. Construct POS circuit from a truth table.

Min and Max Term

In the SOP circuit shown in Fig. 1 above, every product term contains all three input variables. Likewise, in the POS circuit in Fig. 2, every sum term contains all three input variables. Product terms that contain all input variables are known as minterms and sum terms that contain all input variables are known as maxterms. A minterm or maxterm number can be assigned to each row in a truth table if the input 1's and 0's on a given row are interpreted as a binary number. Thus, the SOP equation above (and in the truth table below in Fig. 3) contains minterms 1, 3, and 5, and the POS equation contains maxterms 0, 2, 4, 6, and 7. In an SOP equation, an input value of '1' creates a non-inverted variable in the minterm (and '0' creates an inverted variable). This defines a minterm code that associates each minterm with a corresponding truth table row. In a POS equation, an input value of '1' creates an inverted variable (and '0' creates a non-inverted variable). This defines a maxterm code that associates every maxterm with a particular truth table row.

Figure 3. Derive minterm and maxterm from a truth table.

Using minterm and maxterm codes it is possible to write a new, compact form of SOP and POS equations that follow directly from a truth table. The SOP equation uses the summation symbol $\sum$ to suggest the summing of terms, and the POS equation uses the symbol $\prod$ to suggest taking the product of terms. Both equations simply list the minterms or maxterms present in a given truth table after the initial symbol. Every truth table output row that contains a '1' defines a minterm and every row that contains a '0' defines a maxterm. Minterm and maxterm equations that follow are shown for the truth table above in Fig. 3. SOP $F = \sum m(1, 3, 5)$ and POS $F = \prod M(0, 2, 4, 6, 7)$.

Important Ideas

  • Any logic system can be represented in two logically equivalent ways: as the OR'ing of AND'ed terms, known as the Sum Of Products (SOP) form; or as the AND'ing of OR'ed terms, known as the Product of Sums (POS) form. The two forms are interchangeable, and one form can be transformed to the other following a few basic rules.
  • Product terms that contain all input variables are known as minterms, and Sum terms that contain all input variables are known as maxterms.

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