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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the phrase "up to xxxx" indicates that members of an equivalence class
Equivalence class

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:...
 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 to which it is considered equivalent. In group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
, for example, we may have a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 G acting
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . 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 transformation 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 symmetry of objects using group . 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 transformation of the set....
.

Tetris
A simple example is "there are seven reflecting tetromino
Tetromino

A tetromino, also spelled tetramino or tetrimino, is a geometric shape composed of four square s, connected orthogonality. This is a particular type of polyomino, like dominoes and pentominoes are....
s, up to rotations," which makes reference to the seven possible contiguous arrangements of tetrominoes (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 June 1985, while working for the Dorodnicyn Computing Centre of the Russian Academy of Sciences in Moscow....
 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.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the phrase "up to xxxx" indicates that members of an equivalence class
Equivalence class

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:...
 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 to which it is considered equivalent. In group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
, for example, we may have a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 G acting
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . 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 transformation 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 symmetry of objects using group . 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 transformation of the set....
.

Examples


Tetris


A simple example is "there are seven reflecting tetromino
Tetromino

A tetromino, also spelled tetramino or tetrimino, is a geometric shape composed of four square s, connected orthogonality. This is a particular type of polyomino, like dominoes and pentominoes are....
s, up to rotations," which makes reference to the seven possible contiguous arrangements of tetrominoes (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 June 1985, while working for the Dorodnicyn Computing Centre of the Russian Academy of Sciences in Moscow....
 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 many readers would 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 puzzle
Eight queens puzzle

The eight queens puzzle is the problem of putting eight chess Queen s on an 8?8 chessboard such that none of them is able to capture any other using the standard chess queen's moves....
, 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 several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
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 game of chess, and consists of 64 squares arranged in two alternating colors . The colors are called "black" and "white" , although the actual colors are usually dark green and buff for boards used in competition, and often natural shades of light and dark woods for home boards....
 are occupied by them.

If, in addition to treating the queens as identical, rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
s and reflection
Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
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

The word modulo, in the mathematical community, is often used informally, in many imprecise ways. 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 group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 that "there are two different groups
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of order 4 up to isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
." 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 group s 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

  • Quotient set
  • Synecdoche
    Synecdoche

    Synecdoche is a figure of speech in which:* a term denoting a part of something is used to refer to the whole thing , or* a term denoting a thing is used to refer to part of it , or...