Sum of Products and Product of Sum

SUM OF PRODUCTS AND PRODUCT OF SUM

Logical functions are generally expressed in terms of logical variables values taken on by the logical functions and logical variables are in the binary form. The logical functions can be expressed in two forms they are

i) Sum of products (SOP)

ii) Product of sum (POS)

Product term: The AND function is referred to as a product. The logical product of several variables on which a function depends is considered to be a product term can appear either in complemented (or) uncomplemented form

Sum term: An OR function is generally used to refer a sum. The logical sum of several variables on which a function depends is considers to be a sum term.

Sum of Products (SOP)

The logical sum of two (or) more logical product term is called as sum of product (SOP). If is basically an OR operation of AND operated variable such as

Product of Sum (POS)

A product of sum (POS) is a logical product of two (or) more logical sum terms. It is basically an AND operation of OR operated variables such as

Canonical Form

The canonical forms are the special cases of SOP and POS forms. These are also known us standard SOP and POS forms.

4.10.1. Standard SOP Form

We can realize SOP form, all the individual terms do not involves all the variables an literals. For example, in experts  the second product term do not contain the variable A. If each term in SOP form contain all the variables then term in SOP form is known as standard (or) canonical SOP. Each terms in the standard SOP is called as minterms.

In the above example an the variables are present in the each term. It is the canonical form of SOP.

4.10.2 Standard POS Form

If each term in POS form contains all the variables then POS is known as standard POS form (or) canonical form. The canonical POS form is also known as maxterms example is shown below

Converting Expressions in Standard SOP (or) POS Form

For a three variable experssion with variables A, B and C, if there is a term AB, where C is missing, them we form (C + C) and AND it with AB term.

………..(1)

The equation(1) having all variables A, B, C then is is called canonical SOP (or) standard SOP. Similarly in POS form the missing variable is OR function with the given variables example if the function

The final equation having all the variables, then it is called standard POS (or) canonical POS form.

Problem: 63

Problem: 64

Problem: 65

4.10.3 Minterms and Maxterms

Each individual terms in the standard SOP form is called as minterms and each individual term in standard POS form is called as maxterm. The minterms and maxterms for two variables is shown in the table 4.25 below

Each minterms and maxterm represented by mi and M., where i is the decimal number equivalent of the natural binary number.

Problem: 66

Problem: 67

Problem: 68

Problem: 69