Up to
 |
In mathematics, the phrase up to is useful for modeling fundamental concepts within a realm of mathematical inquiry, and can be compared with the phrase "all other things being equal" in other disciplines. It indicates that its grammatical object is some equivalence class, to be regarded as a single entity, or disregarded as a single entity. If this object is a class of transformations (such as "isomorphism" or "permutation"), it implies the equivalence of objects one of which is the image of the other under such a transformation.
If X is some property or process, a translation of "up to X" is "disregarding a possible difference in X". For instance we might follow the statement "an integer can be factored as a product of prime numbers" with "the product is unique up to ordering", meaning the order of the operands is irrelevant, integers and their prime factorization are; or we might say "the solution to an indefinite integral is f(x), up to addition by a constant", meaning that the constant is not the focus here, the solution f(x) is, and that the addition of a constant is to be regarded as a background, of secondary focus. Further examples concerning up to isomorphism, up to permutations and up to rotations are described below.
In informal contexts, mathematicians often use the word modulo (or simply "mod") for similar purposes, as in "modulo isomorphism".
Examples
Tetris
A simple example is "there are seven reflecting tetrominos, 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) which are frequently thought of as the seven Tetris pieces (box, I, L, J, T, S, Z.) This could also be written "there are five tetrominos, up to reflections and rotations", which would take account of the perspective that L and J could be thought of as the same piece, reflected, as well as that S and Z could be seen as the same. The Tetris game does not allow reflections, so the former notation is likely to seem more natural.
To add in the exhaustive count, there is no formal notation. However, it is common to write "there are seven reflecting tetrominos (=19 total) up to rotations". In this, Tetris provides an excellent example, as a reader might simply count 7 pieces * 4 rotations as 28, where some pieces (box being the obvious example) have fewer than four rotation states.
Eight queens
In the eight queens puzzle, if the eight queens are considered to be distinct, there are 3 709 440 distinct solutions. Normally however, the queens are considered to be identical, and one says "there are 92 (= 3 709 440/8!) unique solutions up to permutations of the queens", or "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, rotations and reflections of the board were allowed, we would have only 12 distinct solutions up to symmetry, signifying that two arrangements that are symmetrical to each other are considered equivalent.
Group Theory
In group theory, for example, we may have a group G acting on a set X, in which case we 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, if we consider groups to be equivalent if they are isomorphic.
See also
