Up to

From Academic Kids

In mathematics, the term "up to xxxx" is used to describe a situation in which members of an equivalence class can 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.

Example: 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.

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.

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.

General usage of this term

In modern slang, "up to" means you are either willing to engage in an act ("Sally is up to going to the park"), capable of an act ("I'm sorry, I'm just not up to it") or are currently engaged in an act. In the latter meaning, its usually phrased in the question 'What are you up to?'. It also sees common use in the terms 'up to no good', indicating someone is engaging in mischief, and 'up to something', indicating someone is engaged in some underhanded activity.

Most generally in English, "up to" means "less than or equal to". In this sense, it is a popular phrase with advertisers and marketing people, as people are expected to respond more positively to "up to 50% off the marked price" than to the logically equivalent "50% or less off the marked price".

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