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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. File links The following pages link to this file: Alchemy Ada Adventure Apartheid Abbreviation Airplane (disambiguation) Abduction Alder Anno Domini Air ABC (disambiguation) Ad hominem Afghan AD Aether Aba Anus Affinity Ai AZ Albinism Accumulator Binary Chess Computer Carbon Cow Cricket (disambiguation) Collection Convex Culture Ceramics Case Creation Crow (disambiguation... Wiktionary full URL is a sister project to Wikipedia intended to be a free wiki dictionary (including thesaurus and lexicon) in every language. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... Group theory is that branch of mathematics concerned with the study of groups. ... The term group can refer to several concepts: Look up Group in Wiktionary, the free dictionary In music, a group is another term for band or other musical ensemble. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, groups are often used to describe symmetries of objects. ...


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. One of the 12 unique solutions The eight queens puzzle is the problem of putting eight chess queens on an 8×8 chessboard such that none of them is able to capture any other using the standard chess queens moves. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ... A chessboard is the board used in the game of chess, which consists of eight rows and eight columns of squares arranged in alternating colors. ...


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. Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. ... The word reflection (also spelt reflexion in British English) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...


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. The word modulo is the Latin ablative of modulus. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...


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. Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...


See also


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Up to - Wikipedia, the free encyclopedia (292 words)
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.
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.
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," 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.
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