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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X "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, 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. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.
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Examples
Tetris
A simple example is "there are seven reflecting tetrominos, 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 pieces (box, I, L, J, T, S, Z. A tetromino, also spelled tetramino or tetrimino is a geometric shape composed of four squares connected orthogonally. Tetris (Тетрис is a Video puzzle game originally designed and programmed by Alexey Pajitnov in June 1985 while working for the Dorodnicyn ) 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, if the eight queens are considered to be distinct, there are 3 709 440 distinct 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 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. In several fields of Mathematics the term permutation is used with different but closely related meanings A chessboard is the type of Checkerboard used in the Game of Chess, and consists of 64 squares (eight rows and eight columns arranged in two alternating
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. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. 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
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. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure " 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. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers
Another typical example is the statement in group theory that "there are two different groups of order 4 up to isomorphism. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective " This means that there are two equivalence classes of groups of order 4, if we consider groups to be equivalent if they are isomorphic. 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
See also
In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X Synecdoche is taken from Greek sinekdohi (συνεκδοχή meaning "simultaneous understanding" (si-nek-duh-kee (pronounced /sɪˈnɛkdoˌki/Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic