Up to

In mathematics, the jargon 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, this may be a group action.

Example: in the eight queens puzzle, if the eight queens are considered to be distinct, there are 3709440 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 chess board 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 2 different groups of order 4 up to isomorphism". This means that there are 2 equivalence classes of groups of order 4, if we consider groups to be equivalent if they are isomorphic.

See also:

• quotient set