# Sum of Product (SOP) and Product of Sum (POS)

**Notation**

Using, everywhere for AND is laborious

We will quietly drop. If the intention is clear

For Example

f = x1'.x2' + x1'.x2 + x1.x2

is better written as

f = x1'x2' + x1'x2 + x1x2

Usual precedence rules of + and . apply

- Do . before + just like in multiplication

- For example 5x + 3y is (5x) + (3y) and not 5(x+3)y

**Some Definition**

A literal is a complemented or uncomplemented boolean variables.

- Examples: a and ā are distinct literals. ā+cd is not.

A Product term is a single literal or a logical product (AND) of two or more literals.

- Examples: a, ā, ac, ācd, aaāb are product terms; ā+cd is not a product term.

A Sum term is a single literal or a logical sum (OR) of two or more literals.

- Examples: a, ā, a+c, ā+c+d are sum terms; ā+cd is not a sum term.

A normal term is a product or sum term in which no variable appears more than once.

- Example: a, ā, a+c, ācd are normal terms; ā+a, āa are not normal terms.

A minterm of n variables is a normal product term with n literals. There are 2 to the power n such product terms.

- Examples of 3-variables minterms: ābc, abc

- Examples: āb is not a 3-variable minterm.

A maxterm of n variable is a normal sum term with n literals. There are 2 to the power n such sum terms.

- Example of 3-variables maxterms: ā+b+c, a+b+c

A sum of product (SOP) expressions is a set of product (AND) terms connected with logical sum (OR) operators.

- Examples: a,ā,ab+c,āc+bde,a+b are SOP expressions.

A product of sum (POS)expressions is a set of sum (OR) terms connected with logical product (AND) operators.

- Examples: a,ā,a+b+c, (ā+c)(b+d) are POS expressions.

A canonical sum of products (CSOP) form of an expression refers to rewriting the expression as a sum of minterms.

- Examples for 3-variables: ābc+abc is a CSOP expression; āb+c is not.

The canonical products of sum (CPOS) form of an expression refers to rewriting the expression as a product of maxterms.

- Examples for 3-variables: (ā+b+c)(a+b+c) is a CPOS expressions; (ā+b)c is not.

There is a close correspondence between the truth table and minterms and maxterms.

**DeMorgan's Theorem (revisited)**

Complement of Sum of Product is equivalent to Product of Complements.

Complement of Product of Sum is equivalent to Sum of Complements.

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