# Understanding Sum of Minterms and Product of Maxterms

A Boolean function is expressed in two form.

1. Sum of Minterms
2. Product of Maxterms

### Sum of Minterms

x’ y’ z , x y’ z’ , x y’ z  , x y z’  , x y z   , gives 1 as output in the above Truth Table.

• Literal –  x, y, A, b etc is a label which denote an input variable for a logic gate. Literal can be normal or complimented.
• Minterm – product of two or more literal using ANDing of each literal.
• Maxterm – sum of two or more literal using ORing of each literal.

Before we understand what sum of minterm or product of maxterm is, we must understand a few terminology.

For example,

``x or x', y or y'``

For example,

``x.y.z or x'y``

Suppose we have 2 variable – x and y, then all possible combination of literals are x’y’ , x’y, xy’, xy. If we have 3 variables then all combination of literals are as follows.

Basically, if there are n variable, then there is 2^n. For 3 variable, there are 2^3 = 8.

A minterm is the term from table given below that gives 1 output.Let us sum all these terms,

``````F = x' y' z + x y' z'  + x y' z + x y z' + x y z

= m1 + m4 + m5 + m6 + m7

F(x,y,z) = ∑(1,4,5,6,7) is known as Sum of Minterms Canonical Form.``````

Why is it called canonical form ? because all the literals present in each of the terms.

### Product of Maxterm

The Product of Maxterm is complement of the Sum of Minterm of a function. To obtain the Product of Maxterm, we need two step process.

1. Find those minterms in the Truth Table that gives a 0 as output.
2. Complement those minterms using DeMorgan’s law.

Let us now apply the above to obtain the Product of Maxterm form.
From the previous truth table given,  x’ y’ z’, x’ y z’, x’ y z gives output as 0.

``````F = x' y' z' + x' y z' + x' y z             by Rule 1
= (x' y' z' + x' y z' + x' y z)'            by DeMorgan's Law
= (x + y + z)(x + y' + z)(x + y' + z')      Product of Maxterms form``````

We see that the Product of Maxterm is ANDing of all ORed terms.