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up to

prep. 1 against; next to; near; towards; as far as. 2 capable of. 3 ready for. 4 willing to participate in. 5 As much as; no more than. 6 until. 7 For the option or decision of. 8 Doing; involved in (with implications of mischief). 9 incumbent; the obligation of; the duty of. 10 (context mathematics English) Considering all members of an equivalence class the same.

up to
  1. adj. busy or occupied with; "what have you been up to?"; "up to no good"

  2. having the requisite qualities for; "equal to the task"; "the work isn't up to the standard I require" [syn: adequate to(p), capable, equal to(p), up to(p)]

Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent. The statement "elements a and b of set S are equivalent up to X" means that a and b are equivalent if criterion X (such as rotation or permutation) is ignored. That is, a and b can be transformed into one another if a transform corresponding to X (rotation, permutation etc.) is applied.

Looking at the entire set S, when X is ignored the elements can be arranged in subsets whose elements are equivalent ("equivalent up to X"). Such subsets are called "equivalence classes".

If X is some property or process, the phrase "up to X" means "disregarding a possible difference in X". For instance the statement "an integer's prime factorization is unique up to ordering", means that the prime factorization is unique if we disregard the order of the factors. We might say "the solution to an indefinite integral is , up to addition by a constant", meaning that the added constant is not the focus here, the solution 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".

Usage examples of "up to".

The water came up to our ankles, and it was as cold as I remembered it.

The trees came up to the road, and the branches blocked out most of the sunlight.

In fact, we had a guy stuck up to his knees in the mud, and a Jeep tried to pull him out, and got sucked in up to the windshield, then a deuce-and-a-half truck tried to pull the guy and the Jeep out, and got buried up to the roof, and then two bulldozers came and they both got buried, then we called in a sky crane chopper with cables, and the chopper got sucked right in and disappeared.

The lobby was small but new, and we took the one elevator up to the rooftop restaurant.

Apparently, Colonel Mang already had it in his mind that I was up to no good, so he had to make what he knew fit with what he suspected.

Now and then I got the motorcycle up to one hundred KPH, and I got good at doing the Vietnamese horn-honking weave.

Dong Ha, but once I cleared them, I got the bike back up to one hundred KPH.

U-turn, and we pulled into the gas station and up to a hand crank pump.

Pine trees came up to the sides of the road, and it was getting spooky.

I got up to eighty KPH, which was much too fast for the road or the bad visibility.

We drove up to the motel office in the middle of a long stucco building and dismounted.

We had come a long way, but beyond that, this moment of truth, which had been abstract up to now, was suddenly real and immediate.

I twisted the throttle and got the bike up to sixty KPH, and I saw that the cloud of dust behind me was getting farther away.

But as we waited on the taxiway, my mind kept pulling me back to 1972, and the events that led up to my second visit to this place.

She seemed to know the alleys and passageways, and within a few minutes, we were pulling up to the parking lot beneath the Nguyen apartment.