Two mathematical objects and are called equal up to an equivalence relation if and are related by , that is, if holds, that is, if the equivalence classes of and with respect to are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, is unique up to means that all objects under consideration are in the same equivalence class with respect to the relation . Moreover, the equivalence relation 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 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 , up to addition of a constant" tacitly employs the equivalence relation between functions, defined by if the difference is a constant function, and means that the solution and the function are equal up to this . In the picture, "there are 4 partitions up to rotation" means that the set has 4 equivalence classes with respect to defined by if can be obtained from 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 " can be understood informally as "ignoring the same subtleties as 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". Examples Tetris A simple example is "there are seven tetrominoes, up to rotations", which makes reference to the seven possible contiguous arrangements of tetrominoes (collections of four unit squares arranged to connect on at least one side) and which are frequently thought of as the seven Tetris pieces (O, I, L, J, T, S, Z). One could also say "there are five tetrominoes, up to reflections and rotations", which would then take into account the perspective that L and J (as well as S and Z)