Representation of Logic Functions - SOP and POS Forms

Representation of Logic Functions - SOP and POS Forms

• Dec.-03, 12, 15

• Boolean expressions are constructed by connecting the Boolean constants and variables with the Boolean operations. These Boolean expressions are also known as Boolean formulas. We use Boolean expressions to describe switching function or Boolean functions. For example, if the Boolean expression  is used to describe the function f, then Boolean function is written as

• Based on the structure of Boolean expression, it can be categorized in different formulas. One such categorization are the normal formulas. Let us consider the four-variable Boolean function.

• In this Boolean function the variables are appeared either in a complemented or an uncomplemented form. Each occurrence of a variable in either a complemented or an uncomplemented form is called a literal. Thus, the above Boolean function 3.2.1 consists of six literals. They appear in the product terms. A product term is defined as either a literal or a product (also called conjunction) of literals. Function 3.2.1 contains three product terms, namely, . Let us consider another four variable Boolean function.

• The above Boolean function consists of seven literals. Here, they appear in the sum terms. A sum term is defined as either a literal or a sum (also called disjunction) of literals. Function 3.2.2 contains three sum terms, namely,  These literals and terms are arranged in one of the two forms :

• Sum of product form (SOP) and

• Product of sum form (POS).

 

1. Sum of Product Form

• The words sum and product are derived from the symbolic representations of the OR and AND functions by + and • (addition and multiplication), respectively. But we realize that these are not arithmetic operators in the usual sense. A product term is any group of literals that are ANDed together. For example, ABC, XY and so on. A sum term is any group of literals that are ORed together such asA + B + C, X + Y and so on. A Sum Of Products (SOP) is a group of product terms ORed together. Some examples of this form are :

• Each of these sum of products expressions consist of two or more product terms (AND) that are ORed together. Each product term consists of one or more literals appearing in either complemented or uncomplemented form. For example, in the sum of products expression the first product term contains literals A, B and C in their uncomplemented form. The second product term contains B and C in their complemented (inverted) form. The sum of product form is also known as disjunctive normal form or disjunctive normal formula.

 

2. Product of Sum Form

• A product of sums is any groups of sum terms

ANDed together. Some examples of this form are :

• Each of these product of sums expressions consist of two or more sum terms (OR) that are ANDed together. Each sum term consists of one or more literals appearing in either complemented or uncomplemented form. The product of sum form is also known as conjunctive normal form or conjunctive normal formula.

 

3. Standard SOP and Standard POS Forms

• We can realize that in the SOP form, all the individual terms do not involve all literals. For example, in expression  the first product term do not contain literal C. If each term in SOP form contains all the literals then the SOP form is known as standard or canonical SOP form. Each individual term in the standard SOP form is called minterm. One standard sum of products expression is as shown in Fig. 3.2.1.

If each term in POS form contains all the literals then the POS form is known as standard or canonical POS form. Each individual term in the standard POS form is called maxterm. One standard product of sums expression is as shown in Fig. 3.2.2.

 

4. Converting Expressions in Standard SOP or POS Form

Stun of product form can be converted to standard sum of products by ANDing the terms in the expression with terms formed by ORing the literal and its complement which are not present in that term. For example for a three literal expression with literals A, B and C, if there is a term AB, where C is missing, then we form term  and AND it with AB. Therefore, we get 

Steps to convert SOP to standard SOP form

Step 1 : Find the missing literal in each product term if any.

Step 2 : AND each product term having missing literal/s with term/s form by ORing the literal and its complement.

Step 3 : Expand the terms by applying distributive law and reorder the literals in the product terms.

Step 4 : Reduce the expression by omitting repeated product terms if any. Because A + A = A.

 

Examples for Understanding

Ex. 3.2.1 Convert the given expression in standard SOP form,

f (A, B, C) = AC + AB + BC

Sol. :

Step 1 : Find the missing literal/s in each product term.

Step 2 :

AND product term with (missing literal + its complement).

Step 3 : Expand the terms and reorder literals.

Note : After having sufficient practice student should expand product term and reorder literals in it in a single step.

Step 4 : Omit repeated product terms.

 

Ex. 3.2.2 Convert the given expression in standard SOP form, f (A, B, C) = A + ABC

Sol. :

Step 1 : Find the missing literal/s in each product term.

Step 2 : AND product term with (missing literal + its complement).

Step 3 : Expand the terms and reorder literals.

Step 4 : Omit repeated product term

 

Example with Solution

Ex. 3.2.3 Define canonical form. Express F = BC' + AC in a canonical SOP form.

Sol. :

 

Examples for Practice

Ex. 3.2.4     Express the following function in standard SOP form

 

Steps to convert POS to standard POS form

Step 1 : Find the missing literals in each sum term if any.

Step 2 : OR each sum term having missing literal/s with term/s form by ANDing the literal and its complement.

Step 3 : Expand the terms by applying distributive law and reorder the literals in the sum terms.

Step 4 : Reduce the expression by omitting repeated sum terms if any. Because A . A = A.

 

Examples for Understanding

Ex. 3.2.6 Convert the given expression in standard POS form.

f (A, B, C) = (A + B) (B + C) (A + C)

Sol. :

Step 1 : Find the missing literal/s in each sum term.

 

Ex. 3.2.7 Convert the given expression in standard POS form. Y = A . ( A + B + C)

Sol. :

Step 1 : Find the missing literal/s in each sum term.

Step 2 : OR sum term with (missing literal • its complement).

Step 3 : Expand the terms and reorder literals.

Since A + BC = (A + B) (A + C) we have,

Step 4 : Omit repeated sum terms.

 

Example with Solution

Ex. 3.2.8 Convert the given expression in standard POS form.

f(A, B, C) = (A + B) . (B + C)

Sol. :

 

Example for Practice

Ex. 3.2.9 Convert the given expression in standard POS form

 

5. M Notations : Minterms and Maxterms

• Each individual term in standard SOP form is called minterm and each individual term in standard POS form is called maxterm. The concept of minterms and maxterms allows us to introduce a very convenient shorthand notations to express logical functions. Table 3.2.1 gives the minterms and maxterms for a three literal/variable logical function where the number of minterms as well as maxterms is 23 = 8. In general, for an n-variable logical function there are 2n minterms and an equal number of maxterms.

• As shown in Table 3.2.1 each minterm is represented by m_i, and each maxterm is represented by M_i, where the subscript i is the decimal number equivalent of the natural binary number. With these shorthand notations logical function can be represented as follows :

where ∑ denotes sum of product while n denotes product of sum.

• We know that logical expression can be represented in the truth table form. It is possible to write logic expression in standard SOP or POS form corresponding to a given truth table. The logic expression corresponding to a given truth table can be written in a standard sum of products form by writing one product term for each input combination that produces an output of 1. These product terms are ORed together to create the standard sum of products. The product terms are expressed by writing complement of a variable when it appears as an input 0, and the variable itself when it appears as an input 1. Consider, for example, the truth Table 3.2.2.

• The product corresponding to input combination 010 is , the product corresponding to input combination 011 is  and product corresponding to input combination 110 is . Thus the standard sum of products form is

• The logic expression corresponding to a truth table can also be written in a standard product of sums form by writing one sum term for each output 0. The sum terms are expressed by writing complement of a variable when it appears as an input 1 and the variable itself when it appears as an input 0. Consider, for example, the truth Table 3.2.3.

The sum corresponding to input combinations 010 is  and the sum corresponding to input 101 is  Thus, the standard product of sum form is

 

6. Complements of Standard Forms

• The POS and SOP functions derived from the same truth table are logically equivalent. In terms of minterms and maxterms we can then write

• From the above expressions we can easily notice that there is a complementary type of relationship between a function expressed in terms of maxterms. Using this complementary relationship we can find logical function in terms of maxterms if function in minterms is known or vice-versa. For example, for a four variables if

f (A, B, C, D) = ∑ m(0, 2, 4,6,8,10,12,14)

then f (A, B, C, D) = π M(l, 3, 5,7,9,11,13,15)

Review Questions

1. Define switching function.

2. Define literal, product term and sum term.

3. Explain sum of product form.

4. What do you mean by standard SOP and POS forms ?

5. Explain how to convert SOP or POS expressions in their standard forms.

6. What do you mean by minterms and maxterms ?

7. Show that a function expressed as a sum of its minterms is equivalent to a function expressed as a product of its maxterms.