Integer Set One Residue Class Membership Tool

Description

Normalize integer modulo positive modulus and return member when the result equals residue, requiring residue from zero through modulus minus one.

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.

Normalize integer modulo positive modulus and return member when the result equals residue, requiring residue from zero through modulus minus one.

When to use this tool

Use Integer Set One Residue Class Membership 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 integer, residue, modulus. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read member. Inspect member before interpreting the associated values.

Normalize integer modulo positive modulus and return member when the result equals residue, requiring residue from zero through modulus minus one.^[1]

Input field	Meaning and units	Accepted input
integer	Integer	Required. Safe integer.
residue	Residue	Required. Safe integer; minimum 0.
modulus	Modulus	Required. Safe integer; minimum 1.

Output field	Interpretation	Returned shape
member	Member	Required. Boolean.

json

{
  "input": {
    "integer": -7,
    "residue": 2,
    "modulus": 3
  },
  "output": {
    "member": true
  }
}

Limitations and assumptions

The input is a safe integer and residue convention is nonnegative. Numeric modulo remains floating-point arithmetic near extreme safe-integer values; this does not enumerate a residue class or provide arbitrary precision.
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

Calculation and input contract: set/integer-set-one-residue-class-membership — tool.io

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