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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 (permuting) the other.

[1] 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.

Objects that are distinct up to an equivalence relation defined by a group action, such as rotation, reflection, or permutation, can be counted using Burnside's lemma or its generalization, Pólya enumeration theorem.

[2]) But if rotations are not considered distinct — so that we treat both "I vertically" and "I horizontally" indifferently as "I" — then there are only seven.

A hyperreal x and its standard part st(x) are equal up to an infinitesimal difference.