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In mathematics
, the phrase "up to x" means "disregarding a possible difference in x".
For instance, when calculating an indefinite integral, one could say that the solution is f(x) "up to addition by a constant," meaning it differs from f(x), if at all, only by some constant.
It indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "x" describes a property or process that transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent. 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
.
s, up to rotations", which makes reference to the seven possible contiguous arrangements of tetrominoes (collections of four unit squares arranged to connect on at least one face) which are frequently thought of as the seven Tetris
pieces (box, I, L, J, T, S, Z.) This could also be written "there are five tetrominos, up to reflections and rotations", which would take account of the perspective that L and J could be thought of as the same piece, reflected, as well as that S and Z could be seen as the same. The Tetris game does not allow reflections, so the former notation is likely to seem more natural.
To add in the exhaustive count, there is no formal notation. However, it is common to write "there are seven reflecting tetrominos (=19 total) up to rotations". In this, Tetris provides an excellent example, as a reader might simply count 7 pieces * 4 rotations as 28, where some pieces (box being the obvious example) have fewer than four rotation states.
, 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 (= 3 709 440/8!) unique solutions up to permutation
s 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, rotation
s and reflection
s 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.
In 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 is an extension of the construct "7 and 11 are equal modulo 4" used in modular arithmetic
, with the assumption that the listener is familiar with such informal mathematical jargon.
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the phrase "up to x" means "disregarding a possible difference in x".
For instance, when calculating an indefinite integral, one could say that the solution is f(x) "up to addition by a constant," meaning it differs from f(x), if at all, only by some constant.
It indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "x" describes a property or process that transforms an element into one from the same equivalence class, i.e. one to which it is considered equivalent. In group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, for example, we may have a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G acting
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
.
Tetris
A simple example is "there are seven reflecting tetrominoTetromino
A tetromino is a geometric shape composed of four squares, connected orthogonally. This, like dominoes and pentominoes, is a particular type of polyomino...
s, up to rotations", which makes reference to the seven possible contiguous arrangements of tetrominoes (collections of four unit squares arranged to connect on at least one face) which are frequently thought of as the seven Tetris
Tetris
Tetris is a puzzle video game originally designed and programmed by Alexey Pajitnov in the Soviet Union. It was released on June 6, 1984, while he was working for the Dorodnicyn Computing Centre of the Academy of Science of the USSR in Moscow, Russian Soviet Federative Socialist Republic...
pieces (box, I, L, J, T, S, Z.) This could also be written "there are five tetrominos, up to reflections and rotations", which would take account of the perspective that L and J could be thought of as the same piece, reflected, as well as that S and Z could be seen as the same. The Tetris game does not allow reflections, so the former notation is likely to seem more natural.
To add in the exhaustive count, there is no formal notation. However, it is common to write "there are seven reflecting tetrominos (=19 total) up to rotations". In this, Tetris provides an excellent example, as a reader might simply count 7 pieces * 4 rotations as 28, where some pieces (box being the obvious example) have fewer than four rotation states.
Eight queens
In the eight queens puzzleEight queens puzzle
The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens attack each other. Thus, a solution requires that no two queens share the same row, column, or diagonal...
, 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 (= 3 709 440/8!) unique solutions up to permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s 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
Chessboard
A chessboard is the type of checkerboard used in the board game chess, and consists of 64 squares arranged in two alternating colors...
are occupied by them.
If, in addition to treating the queens as identical, rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s and reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
s of the board were allowed, we would have only 12 distinct solutions up to symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
, signifying that two arrangements that are symmetrical to each other are considered equivalent.
In informal contexts, mathematicians often use the word modulo
Modulo
In the mathematical community, the word modulo is often used informally. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
(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 is an extension of the construct "7 and 11 are equal modulo 4" used in modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
, with the assumption that the listener is familiar with such informal mathematical jargon.
Another typical example is the statement in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
that "there are two different groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of order 4 up to isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
". 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 isomorphism
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. If there exists an isomorphism between two groups, then the groups are called isomorphic...
.
See also
- ModuloModuloIn the mathematical community, the word modulo is often used informally. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
- Quotient set
- Quotient groupQuotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
- SynecdocheSynecdocheSynecdoche , meaning "simultaneous understanding") is a figure of speech in which a term is used in one of the following ways:* Part of something is used to refer to the whole thing , or...