Everything about Up To totally explained
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. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, for example one to which it's 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.
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.) 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 doesn't allow reflections, so the former notation is likely to seem more natural.
To add in the exhaustive count, there's no formal notation. However, it's 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. 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.
If, in addition to treating the queens as identical,
rotations and
reflections of the board were allowed, we'd 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're
isomorphic.
Further Information
Get more info on 'Up To'.
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