Sum Of Product, Product Of Sum, nSum Of Products & nProduct Of Sums are boolean expressions with multiple inputs.

## 1 (a AND b) OR (c AND d) It’s SOP

## Ex. F = ab’ + ad + c’d + d’

&

2. (a OR b) AND (c OR d) It’s POS

### Ex. F = (a+b’) . (a+d) . (c’+d) . (d’)

### Sum of Product (SOP) Form

The sum-of-products (SOP) form is a method (or form) of simplifying the Boolean expressions of logic gates. In this SOP form of Boolean function representation, the variables are operated by AND (product) to form a product term and all these product terms are ORed (summed or added) together to get the final function.

A sum-of-products form can be formed by adding (or summing) two or more product terms using a Boolean addition operation. Here the product terms are defined by using the AND operation and the sum term is defined by using OR operation.

The sum-of-products form is also called as Disjunctive Normal Form as the product terms are ORed together and Disjunction operation is logical OR. Sum-of-products form is also called as Standard SOP.

SOP form representation is most suitable to use them in FPGA (Field Programmable Gate Arrays).

### Product of Sums (POS) Form

The product of sums form is a method (or form) of simplifying the Boolean expressions of logic gates. In this POS form, all the variables are ORed, i.e. written as sums to form sum terms.

All these sum terms are ANDed (multiplied) together to get the product-of-sum form. This form is exactly opposite to the SOP form. So this can also be said as “Dual of SOP form”.

Here the sum terms are defined by using the OR operation and the product term is defined by using AND operation. When two or more sum terms are multiplied by a Boolean OR operation, the resultant output expression will be in the form of product-of-sums form or POS form.

The product-of-sums form is also called as Conjunctive Normal Form as the sum terms are ANDed together and Conjunction operation is logical AND. Product-of-sums form is also called as Standard POS.

## 1 (a AND b) OR (c AND d) It’s SOP

&

2. (a OR b) AND (c OR d) It’s POS

## • In 1st Case,

**AND is Followed By OR** or

in digital logic, we can say like,

OR is Following AND Operations.

## &

## • In 2nd Case,

## OR is Followed By AND or

## in digital logic, we can say like,

## AND is Following OR Operations.

Two dual canonical forms of *any* Boolean function are a “sum of minterms” and a “product of max terms.” The term “**Sum of Products**” or “**SoP**” is widely used for the canonical form that is a disjunction (OR) of minterms. It’s De Morgans Law is a “**Product of Sums**” or “**PoS**” for the canonical form that is a conjunction (AND) of max terms. These forms can be useful for the simplification of these functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.