#### Boolean algebra is the mathematics used to analyze digital gates and circuits.

Boolean Algebra is made up of the Boolean laws. Maxterm & Minterm are very important topics of Boolean algebra.

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**Minterm**

Minterm is a product of all the literals (in either complement or without complement form).

Example

if we have two boolean variables X and Y

then X.(~Y) is a minterm. we can express complement ~Y as Yâ€™

so, the above minterm can be expressed as XYâ€™

So, if we have two variables then the minterm will consists of product of both the variables

**Maxterm **

Maxterm is a sum of all the literals (with or without complement).

Example

if we have two boolean variables X and Y then X+(Y’) is a maxterm. we can express complement ~Y as Yâ€™ so, the above maxterm can be expressed as X + Yâ€™ So, if we have two variables then the maxterm will consists of sum of both the variables.

**Canonical Expressions **

if we have two variables X and Y then,

Following is a canonical expression consisting of minterms XY + Xâ€™Yâ€™ (Canonical Sum of products-SOP) and

Following is a canonical expression consisting of maxterm (X+Y) . (Xâ€™ + Yâ€™) (Canonical product of Sum POS)

**Different forms of Expressions**

There are two forms of expression. **Sum of Products (SOP)** Â¢This type of expression is formed by adding the product. For example if A and B are two boolean variables. Then Aâ€™B+B is an SOP Â¢In this each term might not have all the variables.

**Product of Sums (POS)** Â¢This type of expression is formed by multiplying the sum. For example if A and B are two boolean variables. Then (Aâ€™+B). B is an SOP Â¢In this each term might not have all the variables.

** Key Differenece between SOP and POS **

â€¢SOP (Sum of product) generates expression in which all the variables in a domain are first multiplied then added. On the contrary, the POS (Product of Sum) represents the Boolean expression having variables summed then multiplied with each other.

â€¢Minterms or product terms are mainly used in the SOP which associates with the high (1) value. Conversely, in POS, Maxterms or sum terms are employed, which produces a low (0) value.

â€¢In the SOP, method, the value â€˜1â€™ is replaced by the variable and â€˜0â€™ by its complement. In contrast, when it comes to POS a â€˜0â€™ is substituted by the variable and â€˜1â€™ by its complement.

â€¢At last, all the terms are added with each other in case of SOP. As against, in POS, the terms are multiplied with each other in the last step of the process.