Editing Up to (section)

[[File:Partitions of 6-set with 3 singletons.svg|thumb|''Top:'' In a [[hexagon]] vertex set there are 20 [[Partition of a set|partitions]] which have one three-element subset (green) and three single-element subsets (uncolored). ''Bottom:'' Of these, there are 4 partitions up to rotation, and 3 partitions up to rotation and reflection.]]{{Short description|Mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)}}Two [[Mathematical object|mathematical objects]] {{mvar|a}} and {{mvar|b}} are called "equal '''up to''' an [[equivalence relation]] {{mvar|R}}"
* if {{mvar|a}} and {{mvar|b}} are related by {{mvar|R}}, that is,
* if {{math|''aRb''}} holds, that is,
* if the [[equivalence classes]] of {{mvar|a}} and {{mvar|b}} with respect to {{mvar|R}} are equal.

This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count.
For example, "{{mvar|x}} is unique up to {{mvar|R}}" means that all objects {{mvar|x}} under consideration are in the same equivalence class with respect to the relation {{mvar|R}}.

Moreover, the equivalence relation {{mvar|R}} is often designated rather implicitly by a generating condition or transformation.
For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation {{mvar|R}} that relates two lists if one can be obtained by reordering ([[permutation|permuting]]) the other.<ref>{{Cite web|url=https://webusers.imj-prg.fr/~jan.nekovar/co/en/en.pdf|title=Mathematical English (a brief summary)|last=Nekovář|first=Jan|date=2011|website=Institut de mathématiques de Jussieu – Paris Rive Gauche|access-date=2024-02-08}}</ref> As another example, the statement "the solution to an indefinite integral is {{math|sin(''x'')}}, up to addition of a constant" tacitly employs the equivalence relation {{mvar|R}} between functions, defined by {{math|''fRg''}} if the difference {{math|''f''−''g''}} is a constant function, and means that the solution and the function {{math|sin(''x'')}} are equal up to this {{mvar|R}}.
In the picture, "there are 4 partitions up to rotation" means that the set {{mvar|P}} has 4 equivalence classes with respect to {{mvar|R}} defined by {{math|''aRb''}} if {{mvar|b}} can be obtained from {{mvar|a}} by rotation; one representative from each class is shown in the bottom left picture part.

Equivalence relations are often used to disregard possible differences of objects, so "up to {{mvar|R}}" can be understood informally as "ignoring the same subtleties as {{mvar|R}} ignores".
In the factorization example, "up to ordering" means "ignoring the particular ordering".

Further examples include "up to isomorphism", "up to permutations", and "up to rotations", which are described in the [[Up to#Examples|Examples section]].

In informal contexts, mathematicians often use the word '''[[modulo (jargon)|modulo]]''' (or simply '''mod''') for similar purposes, as in "modulo isomorphism".

Objects that are distinct up to an equivalence relation defined by a group action, such as rotation, reflection, or permutation, can be counted using [[Burnside's lemma]] or its generalization, [[Pólya enumeration theorem]].