Minterm / Maxterm Expander
Expand a boolean expression into the canonical sum-of-minterms Σm and product-of-maxterms ΠM.
What it does: Writes a boolean expression as the standard sum-of-minterms (Σm) and product-of-maxterms (ΠM).
When to use it: Canonicalizing, comparing Σm/ΠM indices, or preparing standard input for a Karnaugh map or simplification.
Expand into the canonical sum-of-minterms (Σm)? and product-of-maxterms (ΠM)? .
MEANS Variables —: — minterms / — maxterms (complementary indices).
No history yet. Each calculation is automatically saved to this device.
How to use the minterm/maxterm expander
Write an expression → expand → read Σm and ΠM.
- 01
Write a boolean expression
Use single letters A–Z for variables. Operators
AND OR NOT XORor symbols& | ! ^, the trailing primeA', and concatenationABare all supported. - 02
Click "Expand"
The tool first builds the truth table, then lists the minterms where the output is 1 (forming Σm) and the maxterms where the output is 0 (forming ΠM).
- 03
Read both canonical forms
The top shows the index notation Σm(...) / ΠM(...), and below it the fully expanded expression. The two index sets are complementary.
Minterm vs maxterm
The two go in opposite directions and are easy to mix up — use this table to compare.
| Minterm mᵢ | Maxterm Mᵢ | |
|---|---|---|
| Taken from | Rows where output = 1 | Rows where output = 0 |
| Literal joining | AND (product term) | OR (sum term) |
| Variable whose bit is 1 | True variable A | Complemented variable A' |
| Variable whose bit is 0 | Complemented variable A' | True variable A |
| Forms | Σm sum-of-minterms (SOP) | ΠM product-of-maxterms (POS) |
Standard digital logic definitions (e.g. Mano, Digital Design).
Common questions, answered in 3 minutes
What is the difference between a canonical form and a minimal form?
In the canonical form every term contains all variables and it is the unique standard expansion; the minimal form tries to reduce the number of terms and literals. To simplify, use the boolean algebra simplifier or the Karnaugh map tool.
Why are the Σm and ΠM indices complementary?
Each row's output is either 0 or 1: rows with output 1 go into Σm (minterms) and rows with output 0 go into ΠM (maxterms), together covering exactly the full set 0…2ⁿ-1.
How many variables are supported at most?
5, corresponding to up to 32 expansion terms, which keeps the result readable. With more variables the expansion gets very long, so simplify first.
What does the prime A' mean?
The prime denotes negation (NOT). A' means NOT A, matching the notation output by this tool and the simplifier, so you can paste between them directly.
How are always-true / always-false expressions shown?
An always-true expression (tautology) has no maxterms, so ΠM is empty and POS is written as 1; an always-false expression (contradiction) has no minterms, so Σm is empty and SOP is written as 0.
Standards and sources referenced by this tool
| Item | Value / Formula | Source |
|---|---|---|
| Minterm / Σm | sum of minterms | Digital logic canonical SOP |
| Maxterm / ΠM | product of maxterms | Digital logic canonical POS |
Pure algorithmic computation, no external API.