Two mathematical objects ''a'' and ''b'' are called equal up to an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the

equivalence classes In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

of ''a'' and ''b'' with respect to ''R'' are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, ''x'' is unique up to ''R'' means that all objects ''x'' under consideration are in the same equivalence class with respect to the relation ''R''. Moreover, the equivalence relation ''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 ''R'' that relates two lists if one can be obtained by reordering (

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

) from the other. As another example, the statement "the solution to an indefinite integral is sin(''x''), up to addition by a constant" tacitly employs the equivalence relation ''R'' between functions, defined by ''fRg'' if ''f''−''g'' is a constant function, and means that the solution and the function sin(''x'') are equal up to this ''R''. In the picture, "there are 4 partitions up to rotation" means that the set ''P'' has 4 equivalence classes with respect to ''R'' defined by ''aRb'' if ''b'' can be obtained from ''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 ''R''" can be understood informally as "ignoring the same subtleties as ''R'' does". 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 Examples section. In informal contexts, mathematicians often use the word ''

modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...

'' (or simply "mod") for similar purposes, as in "modulo isomorphism".

Examples

Tetris

A simple example is "there are seven reflecting tetrominoes, up to rotations", which makes reference to the seven possible contiguous arrangements of tetrominoes (collections of four unit squares arranged to connect on at least one side) and which are frequently thought of as the seven Tetris pieces (O, I, L, J, T, S, Z). One could also say "there are five tetrominoes, up to reflections and rotations", which would then take into account the perspective that L and J (as well as S and Z) can be thought of as the same piece when reflected. The Tetris game does not allow reflections, so the former statement is likely to seem more relevant. To add in the exhaustive count, there is no formal notation for the number of pieces of tetrominoes. However, it is common to write that "there are seven reflecting tetrominoes (= 19 total) up to rotations". Here, Tetris provides an excellent example, as one might simply count 7 pieces × 4 rotations as 28, but some pieces (such as the 2×2 O) obviously have fewer than four rotation states.

Eight queens

In the eight queens puzzle, if the eight queens are considered to be distinct, then there are 3709440 distinct solutions. Normally, however, the queens are considered to be equal, and one usually says "there are 92 ( unique solutions ''up to''

s of the queens", or that "there are 92 solutions ''modulo'' the names of the queens", signifying that two different arrangements of the queens are considered equivalent if the queens have been permuted, but the same squares on the

chessboard A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the bo ...

are occupied by them. If, in addition to treating the queens as identical,

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

s and reflections of the board were allowed, we would have only 12 distinct solutions ''up to

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

and the naming of the queens'', signifying that two arrangements that are symmetrical to each other are considered equivalent (for more, see ).

Polygons

The regular ''n''-gon, for given ''n'', is unique up to similarity. In other words, if all similar ''n''-gons are considered instances of the same ''n''-gon, then there is only one regular ''n''-gon.

Group theory

group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...

, one may have a group ''G''

acting Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...

on a set ''X'', in which case, one might say that two elements of ''X'' are equivalent "up to the group action"—if they lie in the same

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...

. Another typical example is the statement that "there are two different

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

of order 4 ''up to''

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

", or "''modulo'' isomorphism, there are two groups of order 4". This means that there are two equivalence classes of groups of order 4—assuming that one considers groups to be equivalent if they are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

Nonstandard analysis

A hyperreal ''x'' and its standard part st(''x'') are equal up to an

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...

difference.

Computer science

In computer science, the term ''up-to techniques'' is a precisely defined notion that refers to certain

proof techniques A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof ...

for (weak)

bisimulation In theoretical computer science a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa. Intuitively two systems are bisimilar if ...

, and to relate processes that only behave similarly up to unobservable steps.Damien Pous, ''Up-to techniques for weak bisimulation'', Proc. 32nd ICALP,

Lecture Notes in Computer Science ''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post-proceedings, monographs, and Festschrifts. In addition, tutorial ...

, vol. 3580, Springer Verlag (2005), pp. 730–741