SOP vs POS Explained

Understanding Sum of Products (SOP) and Product of Sums (POS) forms in Boolean algebra with practical examples and conversion methods.

What are SOP and POS?

SOP (Sum of Products)

A Boolean expression where product terms (AND operations) are summed together (OR operations).

F = AB + A'C + BC'

POS (Product of Sums)

A Boolean expression where sum terms (OR operations) are multiplied together (AND operations).

F = (A+B)(A'+C)(B+C')

Key Differences

Aspect SOP POS
Structure Sum of product terms Product of sum terms
Form AB + CD + EF (A+B)(C+D)(E+F)
Minterms/Maxterms Uses minterms (where F=1) Uses maxterms (where F=0)
Implementation AND gates → OR gate OR gates → AND gate

Canonical Forms

Canonical SOP (Minterm Form)

Each product term contains all variables in either normal or complemented form.

F(A,B,C) = A'B'C + A'BC' + AB'C' + ABC

Canonical POS (Maxterm Form)

Each sum term contains all variables in either normal or complemented form.

F(A,B,C) = (A+B+C')(A+B'+C)(A'+B+C)(A'+B'+C')

Truth Table Example

Consider the following truth table for function F(A,B,C):

A B C F Minterm Maxterm
0000-A+B+C
0011A'B'C-
0101A'BC'-
0110-A+B'+C'
1001AB'C'-
1010-A'+B+C'
1101ABC'-
1111ABC-

SOP Expression

Sum minterms where F=1:

F = A'B'C + A'BC' + AB'C' + ABC' + ABC

POS Expression

Product maxterms where F=0:

F = (A+B+C)(A+B'+C')(A'+B+C')

Conversion Between SOP and POS

Method 1: Using De Morgan's Laws

Starting with SOP: F = AB + A'C

  1. 1. Take complement: F' = (AB + A'C)'
  2. 2. Apply De Morgan's: F' = (AB)'(A'C)'
  3. 3. Apply De Morgan's again: F' = (A'+B')(A+C')
  4. 4. Take complement again: F = ((A'+B')(A+C'))'
  5. 5. Final POS: F = (A+B)+(A'+C) = (A+B)(A'+C)

Method 2: Using Truth Table

Convert by identifying minterms (for SOP) or maxterms (for POS) from the truth table.

Applications and When to Use

Use SOP When:

  • • Function has fewer 1s than 0s in truth table
  • • Implementing with AND-OR logic
  • • NAND gate implementation preferred
  • • Easier to identify minterms

Use POS When:

  • • Function has fewer 0s than 1s in truth table
  • • Implementing with OR-AND logic
  • • NOR gate implementation preferred
  • • Easier to identify maxterms

Practical Examples

Example 1: Simple Function

For F(A,B) = A + B:

  • SOP: F = A'B + AB' + AB
  • POS: F = (A + B)

Example 2: XOR Function

For F(A,B) = A ⊕ B:

  • SOP: F = A'B + AB'
  • POS: F = (A + B)(A' + B')

Summary

Both SOP and POS are fundamental forms in Boolean algebra. SOP is often more intuitive and commonly used, while POS can be more efficient for certain functions. Understanding both forms and their conversions is essential for digital logic design and optimization.