Twomathematical objectsa andb are called "equalup to anequivalence relationR"
Thisfigure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count.For example, "x is unique up toR" means that all objectsx under consideration are in the same equivalence class with respect to the relationR.
Moreover, the equivalence relationR is often designated rather implicitly by a generating condition or transformation.
Equivalence relations are often used to disregard possible differences of objects, so "up toR" can be understood informally as "ignoring the same subtleties asR ignores".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 theExamples section.
In informal contexts, mathematicians often use the wordmodulo (or simplymod) for similar purposes, as in "modulo isomorphism".
Objects that are distinct up to an equivalence relation defined by a group action, such as rotation, reflection, or permutation, can be counted usingBurnside's lemma or its generalization,Pólya enumeration theorem.
Consider the sevenTetris pieces (I, J, L, O, S, T, Z), known mathematically as thetetrominoes. If you consider all the possible rotations of these pieces — for example, if you consider the "I" oriented vertically to be distinct from the "I" oriented horizontally — then you find there are 19 distinct possible shapes to be displayed on the screen. (These 19 are the so-called "fixed" tetrominoes.[2]) But if rotations are not considered distinct — so that we treat both "I vertically" and "I horizontally" indifferently as "I" — then there are only seven. We say that "there are seventetrominoes, up to rotation". One could also say that "there are five tetrominoes, up to rotation and reflection", which accounts for the fact that L reflected gives J, and S reflected gives Z.
In theeight queens puzzle, if the queens are considered to be distinct (e.g. if they are colored with eight different colors), then there are 3709440 distinct solutions. Normally, however, the queens are considered to be interchangeable, and one usually says "there are3,709,440 / 8! = 92 unique solutions up topermutation of the queens", or that "there are 92 solutions modulo the names of the queens", signifying that two different arrangements of the queens are considered equivalent if the queens have been permuted, as long as the set of occupied squares remains the same.
If, in addition to treating the queens as identical,rotations andreflections of the board were allowed, we would have only 12 distinct solutions "up tosymmetry and the naming of the queens". For more, seeEight queens puzzle § Solutions.
Theregularn-gon, for a fixedn, is unique up tosimilarity. In other words, the "similarity" equivalence relation over the regularn-gons (for a fixedn) has only one equivalence class; it is impossible to produce two regularn-gons which are not similar to each other.
Ingroup theory, one may have agroupGacting on a setX, in which case, one might say that two elements ofX are equivalent "up to the group action"—if they lie in the sameorbit.
Another typical example is the statement that "there are two differentgroups of order 4 up toisomorphism", or "modulo isomorphism, there are two groups of order 4". This means that, if one considersisomorphic groups "equivalent", there are only two equivalence classes of groups of order 4, prototypically thecyclic group of order 4 and theKlein four-group.
Ahyperrealx and itsstandard partst(x) are equal up to aninfinitesimal difference.
