objects ''a'' and ''b'' are called equal up to an equivalence relation
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes
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
) 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
'' (or simply "mod") for similar purposes, as in "modulo isomorphism".
A simple example is "there are seven reflecting tetrominoe
s, 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.
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 identical, and one usually says "there are 92 ( unique solutions ''up to'' permutation
s of the queens", or that "there are 92 solutions ''mod'' 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
are occupied by them.
If, in addition to treating the queens as identical, rotation
s and reflection
s of the board were allowed, we would have only 12 distinct solutions ''up to symmetry
and the naming of the queens'', signifying that two arrangements that are symmetrical to each other are considered equivalent (for more, see ).
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.
In group theory
, one may have a group
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
Another typical example is the statement that "there are two different groups
of order 4 ''up to'' isomorphism
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
''x'' and its standard part
st(''x'') are equal up to an infinitesimal
In computer science, the term ''up-to techniques'' is a precisely defined notion that refers to certain proof techniques
for (weak) bisimulation
, 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, vol. 3580, Springer Verlag (2005), pp. 730–741]
*Abuse of notation
* All other things being equal
*List of mathematical jargon
* Quotient set
Up-to Techniques for Weak Bisimulation