Finite Relation Injectivity Audit
Audit finite relation left-uniqueness (relational injectivity); functionality does not require totality.
Description
Audit finite relation left-uniqueness (relational injectivity); functionality does not require totality.
An injectivity-style relation check asks whether any codomain element has more than one distinct preimage.
When to use this tool
Audit finite relation left-uniqueness (relational injectivity); functionality does not require totality.
- Prepare the supplied pairsJson, domain, codomain 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 pairsJson, domain, codomain. 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.holds, result.witnesses. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Validate domain/codomain membership, deduplicate pairs and list codomain labels reached from multiple distinct domain labels. holds means no such witness.1
| Input field | Meaning and representation | Accepted input |
|---|---|---|
| pairsJson | JSON text, entered as one text field. The decoded structure is illustrated below; it is not a separate file. | Text |
| domain | domain. | List of strings; at most 128 entries |
| codomain | codomain. | List of strings; at most 128 entries |
For the JSON text fields, the following decoded structures show the example records before they are serialized into input strings. Keep field names, list nesting and value types as shown.
[
{
"field": "pairsJson",
"decoded": [
[
"a",
"x"
],
[
"b",
"y"
]
]
}
]| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.holds | True exactly when the stated property passes; it does not establish other properties. |
| result.witnesses | Concrete missing or conflicting relation evidence; inspect holds alongside the list. |
{
"input": {
"pairsJson": "[[\"a\",\"x\"],[\"b\",\"y\"]]",
"domain": [
"a",
"b"
],
"codomain": [
"x",
"y"
]
},
"output": {
"result": {
"holds": true,
"witnesses": []
}
}
}Limitations and assumptions
- The relation need not be a total or single-valued function. This check alone does not establish a bijection.
- 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 Composition Pairs
Compose finite relations by following first then second, coalescing duplicate witnesses into unique endpoint pairs; order convention is explicit.
- Finite Relation Converse Pairs
Form the converse of a supplied finite homogeneous relation, coalescing duplicate input pairs but retaining first-appearance pair order.