In mathematics, the jargon term "up to xxxx" is used to describe a situation in which members of an equivalence class can 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 which is considered equivalent to it. In group theory, for example, this may be a group action.
Example: in the eight queens puzzle, if the eight queens are considered to be distinct, there are 3709440 distinct solutions. Normally however, the queens are considered to be identical, and one says "there are 92 (= 3709440/8!) 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 chess board 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, signifying that two arrangements that are symmetrical to each other are considered equivalent.
Another typical example is the statement in group theory that "there are 2 different groups of order 4 up to isomorphism". This means that there are 2 equivalence classes of groups of order 4, if we consider groups to be equivalent if they are isomorphic.
See also:
Common misspelling and questions (FAQ)
p-to u-to upto up-o up-t pu-to u-pto upt-o up-ot up-t uup-to upp-to up--to up-tto up-too 7p-to yp-to hp-to 8p-to jp-to 8p-to ip-to jp-to u0-to uo-to ul-to u--to u;-to u--to u[-to u;-to up0to uppto up[to up-5o up-ro up-fo up-6o up-go up-6o up-yo up-go up-t9 up-ti up-tk up-t0 up-tl up-t0 up-tp up-tl up-yo up-tyo up-tosWell, for two of them, I know them to bee as if he fight longer then he sees reason, Ile forswear Armes. that this fat Rogue will tell vs, when we meete at Supper: blowes, what extremities he endured; and in the reproofe necessary, and meete me to morrow night in Eastcheape, The vnyoak'd humor of your idlenesse: Who doth permit the base contagious cloudes That when he please againe to be himselfe, By breaking through the foule and vgly mists If all the yeare were playing holidaies, But when they seldome come, they wisht-for come, So when this loose behauiour I throw off, By how much better then my word I am, And like bright Mettall on a sullen ground: Shall shew more goodly, and attract more eyes, Ile so offend, to make offence a skill, Blunt, Vnapt to stirre at these indignities, You tread vpon my patience: But be sure, Mighty, and to be fear'd, then my condition And therefore lost that Title of respect, The scourge of greatnesse to be vsed on it, Haue holpe to make so portly Nor. My Lord King. Worcester get thee gone: for I do see O sir, your presence is too bold and peremptory, The moody Frontier of a seruant brow, Your vse and counsell, we shall send for.