📘 Introduction to SOP and POS
In Boolean Algebra, we often express logic functions in standard formats for easier circuit design.
There are two main standard forms:
- SOP (Sum of Products) Form
- POS (Product of Sums) Form
Both are ways of organizing logic expressions based on AND, OR, and NOT operations.
🧠 What is SOP (Sum of Products)?
- Definition:
SOP means adding (ORing) multiple ANDed terms. - Structure:
Each term is a product (AND operation) of variables (either normal or complemented), and then all these terms are summed (ORed). - Format: (A⋅B⋅C)+(A⋅B‾⋅C)+(A‾⋅B⋅C‾)(A \cdot B \cdot C) + (A \cdot \overline{B} \cdot C) + (\overline{A} \cdot B \cdot \overline{C})(A⋅B⋅C)+(A⋅B⋅C)+(A⋅B⋅C)
- Simple Words:
AND first, then OR the results. - Example:
- Expression: F=A⋅B+A‾⋅CF = A \cdot B + \overline{A} \cdot CF=A⋅B+A⋅C
- Meaning:
- First do AAA AND BBB,
- Then do A‾\overline{A}A AND CCC,
- Finally OR the two results.
🔥 Steps to Write SOP Form
- Find all input conditions where output = 1.
- Write a product term (AND) for each condition.
- OR (+) all the product terms together.
🧠 What is POS (Product of Sums)?
- Definition:
POS means multiplying (ANDing) multiple ORed terms. - Structure:
Each term is a sum (OR operation) of variables (normal or complemented), and then all these terms are multiplied (ANDed). - Format: (A+B+C‾)⋅(A‾+C)(A + B + \overline{C}) \cdot (\overline{A} + C)(A+B+C)⋅(A+C)
- Simple Words:
OR first, then AND the results. - Example:
- Expression: F=(A+B‾)⋅(B+C)F = (A + \overline{B}) \cdot (B + C)F=(A+B)⋅(B+C)
- Meaning:
- First do AAA OR B‾\overline{B}B,
- Then do BBB OR CCC,
- Finally AND the two results.
🔥 Steps to Write POS Form
- Find all input conditions where output = 0.
- Write a sum term (OR) for each condition.
- AND (·) all the sum terms together.
📊 Comparison Table: SOP vs POS
| Aspect | SOP (Sum of Products) | POS (Product of Sums) |
|---|---|---|
| Operation | OR of AND terms | AND of OR terms |
| Used for | Output = 1 conditions | Output = 0 conditions |
| Example | A⋅B+A‾⋅CA \cdot B + \overline{A} \cdot CA⋅B+A⋅C | (A+B)⋅(B+C‾)(A+B) \cdot (B+\overline{C})(A+B)⋅(B+C) |
| Circuit Type | OR-AND Circuit | AND-OR Circuit |
✏️ Examples
Example 1: SOP Form
Given the truth table:
| A | B | Output F |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
- Output is 1 when:
- A = 0, B = 1 → A‾⋅B\overline{A} \cdot BA⋅B
- A = 1, B = 0 → A⋅B‾A \cdot \overline{B}A⋅B
- SOP expression:
F=A‾⋅B+A⋅B‾F = \overline{A} \cdot B + A \cdot \overline{B}F=A⋅B+A⋅B
(This is actually the XOR operation!)
Example 2: POS Form
Given the same truth table:
| A | B | Output F |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
- Output is 0 when:
- A = 0, B = 0 → (A+B)(A+B)(A+B)
- A = 1, B = 1 → (A‾+B‾)(\overline{A}+\overline{B})(A+B)
- POS expression:
F=(A+B)⋅(A‾+B‾)F = (A+B) \cdot (\overline{A}+\overline{B})F=(A+B)⋅(A+B)
🛠 Canonical SOP and POS
Sometimes you will hear about Canonical (Standard) forms.
- Canonical SOP: Each product term includes all input variables (either normal or complemented).
- Canonical POS: Each sum term includes all input variables.
🚀 Summary
| Concept | SOP | POS |
|---|---|---|
| Full Form | Sum of Products | Product of Sums |
| Operations | First AND, then OR | First OR, then AND |
| Used when | Output = 1 | Output = 0 |
| Shortcuts | Minterm Form | Maxterm Form |
- SOP = Best when focusing on when output is 1.
- POS = Best when focusing on when output is 0.
Both forms are essential in circuit design and logic simplification!