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Concept
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Summary
Two mathematical objects a and b are called equal up to an equivalence relation R
if a and b are related by R, that is,
if aRb holds, that is,
if the equivalence classes of a and b with respect to R are equal.
This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count.
For example, x is unique up to R means that all objects x under consideration are in the same equivalence class with respect to the relation R.
Moreover, the equivalence relation R is often designated rather implicitly by a generating condition or transformation.
For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation R that relates two lists if one can be obtained by reordering (permutation) from the other. As another example, the statement "the solution to an indefinite integral is sin(x), up to addition of a constant" tacitly employs the equivalence relation R between functions, defined by fRg if the difference f−g is a constant function, and means that the solution and the function sin(x) are equal up to this R.
In the picture, "there are 4 partitions up to rotation" means that the set P has 4 equivalence classes with respect to R defined by aRb if b can be obtained from a by rotation; one representative from each class is shown in the bottom left picture part.
Equivalence relations are often used to disregard possible differences of objects, so "up to R" can be understood informally as "ignoring the same subtleties as R ignores".
In the factorization example, "up to ordering" means "ignoring the particular ordering".
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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