Finite Relation Domain and Image
Return finite homogeneous relation domain and image in explicit universe order; preserve isolated universe elements as absent, not inferred pairs.
Description
Return finite homogeneous relation domain and image in explicit universe order; preserve isolated universe elements as absent, not inferred pairs.
The domain of a relation consists of first components, while its image consists of second components actually reached.
When to use this tool
Return finite homogeneous relation domain and image in explicit universe order; preserve isolated universe elements as absent, not inferred pairs.
- Prepare the supplied pairsJson, 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 pairsJson, 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.domain, result.image. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.
Validate pairs in the unique universe and return each universe element appearing on the left as domain and on the right as image.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 |
| universe | universe. | 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",
"b"
],
[
"b",
"a"
]
]
}
]| Output field | Interpretation |
|---|---|
| result | Object containing the named outputs below. |
| result.domain | The domain list in the ordering and representation described above. |
| result.image | The image list in the ordering and representation described above. |
{
"input": {
"pairsJson": "[[\"a\",\"b\"],[\"b\",\"a\"]]",
"universe": [
"a",
"b"
]
},
"output": {
"result": {
"domain": [
"a",
"b"
],
"image": [
"a",
"b"
]
}
}
}Limitations and assumptions
- Lists retain universe order. Image is actual coverage rather than the entire declared universe, and no total-function condition is required.
- 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.