Up to
Template:Short descriptionTwo mathematical objects Template:Mvar and Template:Mvar are called "equal up to an equivalence relation Template:Mvar"
- if Template:Mvar and Template:Mvar are related by Template:Mvar, that is,
- if Template:Math holds, that is,
- if the equivalence classes of Template:Mvar and Template:Mvar with respect to Template:Mvar are equal.
This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, "Template:Mvar is unique up to Template:Mvar" means that all objects Template:Mvar under consideration are in the same equivalence class with respect to the relation Template:Mvar.
Moreover, the equivalence relation Template:Mvar is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation Template:Mvar that relates two lists if one can be obtained by reordering (permuting) the other.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> As another example, the statement "the solution to an indefinite integral is Template:Math, up to addition of a constant" tacitly employs the equivalence relation Template:Mvar between functions, defined by Template:Math if the difference Template:Math is a constant function, and means that the solution and the function Template:Math are equal up to this Template:Mvar. In the picture, "there are 4 partitions up to rotation" means that the set Template:Mvar has 4 equivalence classes with respect to Template:Mvar defined by Template:Math if Template:Mvar can be obtained from Template:Mvar by rotation; one representative from each class is shown in the bottom left picture part.
Equivalence relations are often used to disregard possible differences of objects, so "up to Template:Mvar" can be understood informally as "ignoring the same subtleties as Template:Mvar 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 the Examples section.
In informal contexts, mathematicians often use the word modulo (or simply mod) 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 using Burnside's lemma or its generalization, Pólya enumeration theorem.
ExamplesEdit
TetrisEdit
Consider the seven Tetris pieces (I, J, L, O, S, T, Z), known mathematically as the tetrominoes. 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.<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Tetromino%7CTetromino.html}} |title = Tetromino |author = Weisstein, Eric W. |website = MathWorld |access-date = 2023-09-26 |ref = Template:SfnRef }}</ref>) 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 seven tetrominoes, 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.
Eight queensEdit
In the eight 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 are Template:Math unique solutions up to permutation 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 and reflections of the board were allowed, we would have only 12 distinct solutions "up to symmetry and the naming of the queens". For more, see Template:Format link.
PolygonsEdit
The [[regular polygon|regular Template:Mvar-gon]], for a fixed Template:Mvar, is unique up to similarity. In other words, the "similarity" equivalence relation over the regular Template:Mvar-gons (for a fixed Template:Mvar) has only one equivalence class; it is impossible to produce two regular Template:Mvar-gons which are not similar to each other.
Group theoryEdit
In group theory, one may have a group Template:Mvar acting on a set Template:Mvar, in which case, one might say that two elements of Template:Mvar are equivalent "up to the group action"—if they lie in the same orbit.
Another typical example is the statement that "there are two different groups of order 4 up to isomorphism", or "modulo isomorphism, there are two groups of order 4". This means that, if one considers isomorphic groups "equivalent", there are only two equivalence classes of groups of order 4.
Nonstandard analysisEdit
A hyperreal Template:Mvar and its standard part Template:Math are equal up to an infinitesimal difference.
See alsoEdit
- Abuse of notation
- Adequality
- Essentially unique
- List of mathematical jargon
- Modulo (jargon)
- Quotient group
- Quotient set
ReferencesEdit
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