Canonical Sum of Product
Canonical Sum of Products (SOP) is a standard form of Boolean expression in which all product terms are connected using the OR operation. Each product term contains every input variable exactly once, either in its complemented or uncomplemented form.
Canonical SOP is also called the Sum of Minterms because each product term represents a minterm. It is commonly used in digital logic design to represent Boolean functions systematically.
Minterm in SOP
A minterm is a product term in Boolean algebra that contains every input variable exactly once, either in complemented or uncomplemented form.
For two variables A and B, the minterms are:
| A | B | Minterm | Notation |
|---|---|---|---|
| 0 | 0 | (A’B’) | (m_0) |
| 0 | 1 | \(A’B\) | \(m_1\) |
| 1 | 0 | \(AB’\) | \(m_2\) |
| 1 | 1 | \(AB\) | \(m_3\) |
Rules for writing a minterm:
-
If a variable has value 0, write it in complemented form (A′).
-
If a variable has value 1, write it in uncomplemented form (A).
Example: When \(A=0\) and \(B=1\), the minterm is \(A’B\).
Canonical SOP Example – 2 Variables
Consider the Boolean function with two variables (also called literals), A and B:
F(A, B) = A’B + AB
This expression is in canonical SOP form because each product term contains both variables, A and B.
- A’B: A is complemented, and B is uncomplemented.
- AB: Both A and B are uncomplemented.
The product terms are connected by the OR operation (+), so the expression follows the canonical SOP form.
Canonical SOP from a Truth Table
A Boolean function can be converted into canonical SOP form using its truth table. The process involves selecting the rows where the output is 1 and writing the corresponding minterms.
For example, consider the following truth table:
| A | B | F |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
The output is 1 for the input combinations (0, 1) and (1, 1).
- For A = 0 and B = 1, the minterm is A’B.
- For A = 1 and B = 1, the minterm is AB.
Therefore, the canonical SOP expression is:
F(A, B) = A’B + AB
2 Variables Canonical SOP – Logic Circuit Diagram
Rules for Writing Canonical SOP
Follow these rules to write a Boolean function in canonical SOP form:
- Identify all truth-table rows where the output is 1.
- Write one product term for each selected row.
- Include every input variable in each product term.
- Use a complemented variable when its input value is 0.
- Use an uncomplemented variable when its input value is 1.
- Connect all product terms using the OR operation.
Canonical SOP Notation
Canonical SOP can also be represented using minterm notation.
For example:
F(A, B) = A’B + AB = Σm(1, 3) = Σ m1 + m3
Here, Σ represents the sum of minterms, while the numbers 1 and 3 indicate the truth-table row indices where the function output is 1, using the variable order A, B.
Difference Between Canonical SOP and SOP
Canonical SOP is a special type of SOP in which every product term contains all input variables.
| Feature | Canonical Sum of Products | Sum of Products (SOP) |
|---|---|---|
| Definition | A Boolean expression in which every product term contains all input variables. | A Boolean expression in which product terms are connected using the OR operation. |
| Product Terms | Every term contains all variables exactly once. | Terms may contain all or only some variables. |
| Type of Terms | Each product term is a minterm. | Product terms are not necessarily minterms. |
| Complemented Variables | Each variable appears in complemented or uncomplemented form in every term. | Variables may be complemented, uncomplemented, or absent from a term. |
| Derivation | Obtained directly from truth-table rows where the output is 1. | Can be obtained directly or by simplifying a Boolean expression. |
| Example | (A’B+AB) | (A+B) |
| Other Name | Sum of Minterms. | General Sum of Products form. |
Applications of Canonical SOP
Canonical SOP is useful in digital logic design for the following purposes:
- Boolean Function Representation: It provides a systematic way to represent Boolean functions.
- Logic Circuit Design: It helps implement Boolean functions using AND and OR gates, with NOT gates where required.
- Truth Table Conversion: It converts truth-table outputs into Boolean expressions.
- Logic Function Comparison: It provides a standard form for comparing Boolean functions.
- Boolean Simplification: It provides a starting point for simplifying expressions using Boolean algebra or Karnaugh maps (K-maps).
Conclusion
Canonical Sum of Products (SOP) is a standard Boolean expression in which every product term contains all input variables exactly once. It is constructed by selecting the truth-table rows where the output is 1 and combining their minterms using the OR operation. Canonical SOP is an important concept in Boolean algebra and digital logic design.