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In mathematics, the phrase up to is used to convey the idea that some objects in the same class — while distinct — may be considered to be equivalent under some condition or transformation. It often appears in discussions about the elements of a set, and the conditions under which some of those elements may be considered to be equivalent. More specifically, given two elements , " and are equivalent up to " means that and are equivalent, if criterion , such as rotation or permutation, is ignored. In which case, the elements of can be arranged in subsets known as "equivalence classes", sets whose elements are equivalent to each other up to . In some cases, this might mean that and can be transformed into one another—if a transformation corresponding to (e.g., rotation, permutation) is

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• Up to
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• In mathematics, the phrase up to is used to convey the idea that some objects in the same class — while distinct — may be considered to be equivalent under some condition or transformation. It often appears in discussions about the elements of a set, and the conditions under which some of those elements may be considered to be equivalent. More specifically, given two elements , " and are equivalent up to " means that and are equivalent, if criterion , such as rotation or permutation, is ignored. In which case, the elements of can be arranged in subsets known as "equivalence classes", sets whose elements are equivalent to each other up to . In some cases, this might mean that and can be transformed into one another—if a transformation corresponding to (e.g., rotation, permutation) is
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• In mathematics, the phrase up to is used to convey the idea that some objects in the same class — while distinct — may be considered to be equivalent under some condition or transformation. It often appears in discussions about the elements of a set, and the conditions under which some of those elements may be considered to be equivalent. More specifically, given two elements , " and are equivalent up to " means that and are equivalent, if criterion , such as rotation or permutation, is ignored. In which case, the elements of can be arranged in subsets known as "equivalence classes", sets whose elements are equivalent to each other up to . In some cases, this might mean that and can be transformed into one another—if a transformation corresponding to (e.g., rotation, permutation) is applied. If is some property or process, then the phrase "up to " can be taken to mean "disregarding a possible difference in ". For instance, the statement "an integer's prime factorization is unique up to ordering" means that the prime factorization is unique—when we disregard the order of the factors. One might also say "the solution to an indefinite integral is , up to addition by a constant", meaning that the focus is on the solution rather than the added constant, and that the addition of a constant is to be regarded as a background information. Further examples include "up to isomorphism", "up to permutations" and "up to rotations", which are described in the Examples section. In informal contexts, mathematicians often use the word modulo (or simply "mod") for similar purposes, as in "modulo isomorphism".
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