Description
Check cardinality balance of supplied partition block sizes without claiming their disjointness.
A finite set records membership, while a multiset also records multiplicity. An ordered set adds an ordering rule, and an interval adds endpoint-closure choices. A witness for one element, a count budget, and equality of complete sets establish different facts.
Check cardinality balance of supplied partition block sizes without claiming their disjointness.
When to use this tool
Use Partition Block Size Residual when you need the stated finite-set, interval, multiplicity, membership, or ordering operation under the given equality convention.
How to use the tool and how it works
- Enter blockSizes, universeSize. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read assignedDecimal, residualDecimal, matchesSize. Inspect matchesSize before interpreting the associated values.
Sum nonnegative integer blockSizes with BigInt and subtract from universeSize, returning exact decimal assigned and residual plus matchesSize.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| blockSizes | Block Sizes | Required. Array of safe integer; minimum 0; maximum items 256. |
| universeSize | Universe Size | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| assignedDecimal | Assigned Decimal | Required. Text; maximum UTF-16 length 8192. |
| residualDecimal | Residual Decimal | Required. Text; maximum UTF-16 length 8192. |
| matchesSize | Matches Size | Required. Boolean. |
{
"input": {
"blockSizes": [
2,
3
],
"universeSize": 6
},
"output": {
"assignedDecimal": "5",
"residualDecimal": "1",
"matchesSize": false
}
}Limitations and assumptions
- Equal totals do not prove that blocks are disjoint or cover the intended universe. Negative residual is reported, and no members or block identities are examined.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Use a full set operation when one element or count cannot establish the required relation. Preserve the distinction between strings, typed scalar atoms, nested sets, ordered lists, and multisets.
References
Similar or alternative tools
- Partition Block Count Change
Compare declared partition block counts without constructing or validating the partitions.
- Set Partition Block Disjointness Pair
Return distinct shared witnesses in first-list order and disjoint when that witness list is empty.