A Boolean function is expressed in two form.

- Sum of Minterms
- Product of Maxterms

**Sum of Minterms**

**gives 1 as output in the above Truth Table.**

x’ y’ z , x y’ z’ , x y’ z, x y z’, x y z

x’ y’ z , x y’ z’ , x y’ z, x y z’, x y z

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 followsBasically, if there are

**n**variable, then there is**2**^{n}. For 3 variable, there are**2**^{3 }= 8.A

F =

=

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.

*minterm*is the term from above table 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**=

**m**_{1}+ m_{4}+ m_{5}+ m_{6}+ m_{7}_{}**F(x,y,z) = ∑(1,4,5,6,7) is known as Sum of Minterms Canonical Form.****Why 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.

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

Let us now apply the above to obtain the Product of Maxterm form.

In the above Truth Table, **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.**