# Product of sums Expression

As name suggest, A product of sums expression contains the product of different sum terms, now as like sum of products expression, each term may be either a single literal or a summation of more than one literal. Now take a simple example. Let A, B and C is the different literals of a binary system. Now those literals may connected each other in different combination (maximum 8 combination) by the relation of product with each other’s like A+B+C, A+B+C^, A+B+^C, A+B^+C^, A^+B+C, A^+B+C^, A^+B^+C, A^+B^+C^. So out of those input combination creating by literals some of produce logic ‘0’ at the output. It can be easily obtained from the truth table directly by considering those input combinations that produce logic ‘0’ at the output. Each such input combination which produces logic 0 at output is called a term. Now any expression where the different terms are interrelated by products with each other is called Product of sums expression. The important thing to remember that Here, ‘0’ and ‘1’ respectively mean the un-complemented and complemented variables, unlike sum-of-products expressions where ‘0’ and ‘1’ respectively mean complemented and un-complemented variables.

Let take an example of product of sums expression of three literal A, B and C. Now see the truth table of A, B and C literal for any digital system.

From this above truth table we can get the sum of products expression is

Y = (A+B+C^) . (A+B^+C) . (A^+B+C) . (A^+B^+C^).

Considering the first term, the output is ‘0’ when A = 0, B = 0 and C = 1. This is possible only when A, B and C^ are ORed. Also, for the second term, the output is ‘0’ only when B^, C and A are ORed.

Other terms can be explained similarly. A product of sum expression is also known as a *maxterm Expression.*