Finite Relation Membership Pair List
Decode a square binary finite relation membership matrix into unique ordered pairs with explicit universe labels.
Description
Decode a square binary finite relation membership matrix into unique ordered pairs with explicit universe labels.
A binary membership matrix can encode a relation on an ordered universe. Row and column order determine pair orientation.
When to use this tool
Decode a square binary finite relation membership matrix into unique ordered pairs with explicit universe labels.
- Prepare the supplied matrix, universe fields for this operation.
- Compare the complete example with the representation used by your own input before carrying the result into another step.
How to use the tool and how it works
- Enter matrix, universe. Use the exact text syntax, units and index conventions explained below.
- Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
- Read result.pairs. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Require a unique universe and a square matrix of the same size containing only zero or one. For every one at row j, column k emit [universe[j], universe[k]] in row-major order.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| matrix | matrix. | Numeric rows; at most 128 rows and 128 values per row |
| universe | universe. | List of strings; at most 128 entries |
| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.pairs | The pairs list in the ordering and representation described above. |
{
"input": {
"matrix": [
[
0,
1
],
[
1,
0
]
],
"universe": [
"a",
"b"
]
},
"output": {
"result": {
"pairs": [
[
"a",
"b"
],
[
"b",
"a"
]
]
}
}
}Limitations and assumptions
- No symmetry or other relation property is assumed. Nonbinary or incorrectly sized matrices are input errors.
- All declared input fields are required and unexpected top-level fields are rejected. Numbers must be finite; lists are limited to 128 entries and ordinary text fields to 8,192 UTF-16 code units, with tighter limits where stated.
- JSON text inputs must parse within bounded nesting and size limits. Numeric outputs outside the safe finite range are rejected; large exact integer results use decimal strings. Indexes and half-open spans are zero-based unless a rank is explicitly described as one-based.
Alternative or Complementary methods
Declare the full universe and distinguish testing a property from adding its closure. Use separate functionality and totality checks before interpreting a relation as a function.
References
Similar or alternative tools
- Finite Relation Pair Membership Table
Convert an explicit finite homogeneous relation to a binary membership matrix with caller universe order; duplicate pairs coalesce.
- Finite Relation Antisymmetric Pair Audit
Audit finite relation antisymmetry and return explicit failing witnesses; unknown universe elements are rejected.