Up to

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This article is about the sense of "up to" used in mathematics. See "up to" in Wiktionary for other definitions.

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In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one which is considered equivalent to it. 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.

Examples

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 (= 3709440/8!) unique solutions up to permutations 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.

In very informal contexts, mathematicians often use the word modulo (or simply "mod") for the same purpose, as in "modulo isomorphism, there are two groups of order 4," or "there are 92 solutions mod the names of the queens." This a deliberate misuse of the word "modulo" from modular arithmetic (which also relates to partitioning into equivalence sets), with the assumption that the listener is mathematically sophisticated enough to be in on the joke.

Another typical example is the statement in group theory that "there are two different groups of order 4 up to isomorphism." 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