World Library  
Flag as Inappropriate
Email this Article

Up to

Article Id: WHEBN0000044787
Reproduction Date:

Title: Up to  
Author: World Heritage Encyclopedia
Language: English
Subject: List of small groups, Characterization (mathematics), Algebraic closure, Graeco-Latin square, 2-group
Collection: Mathematical Terminology
Publisher: World Heritage Encyclopedia

Up to

20 partitions of a 6-set have 3 singletons. Up to rotation there are 4, up to rotation and reflection 3.

In mathematics, the phrase up to indicates that its grammatical object is some equivalence class, to be regarded as a single entity, or disregarded as a single entity. If this object is a class of transformations (such as "isomorphism" or "permutation"), it implies the equivalence of objects one of which is the image of the other under such a transformation.

If X is some property or process, the phrase "up to X" means "disregarding a possible difference in X". For instance we might follow the statement "an integer's prime factorization is unique up to ordering", meaning that the prime factorization is unique if we disregard the order of the factors; or we might say "the solution to an indefinite integral is f(x), up to addition by a constant", meaning that the added constant is not the focus here, the solution f(x) is, and that the addition of a constant is to be regarded as a background, of secondary focus. Further examples concerning up to isomorphism, up to permutations and up to rotations are described below.

In informal contexts, mathematicians often use the word modulo (or simply "mod") for similar purposes, as in "modulo isomorphism".


  • Examples 1
    • Tetris 1.1
    • Eight queens 1.2
    • Group theory 1.3
  • Non-standard analysis 2
  • See also 3



Tetris pieces I,J,L,O, S,T,Z

A simple example is "there are seven reflecting tetrominos, 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 side) which are frequently thought of as the seven Tetris pieces (O, 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 (the 2×2 O being the obvious example) have fewer than four rotation states.

Eight queens

A solution of the eight queens problem

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 (= \tfrac{3 709 440}{8!}) unique solutions up to permutations of the queens", or "there are 92 solutions mod the names 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, rotations and reflections of the board were allowed, we would have only 12 distinct solutions up to symmetry and the naming of the queens, signifying that two arrangements that are symmetrical to each other are considered equivalent.

Group theory

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.

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 there are two equivalence classes of groups of order 4, assuming we consider groups to be equivalent if they are isomorphic.

Non-standard analysis

A hyperreal x and its standard part st(x) are equal up to an infinitesimal difference.

See also

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.