# “Up to”

• A figure of speech used in mathematics in connection with expressions derived from equality
• E.g. uniqueness or count
• $$x$$ is unique up to $$R$$” means that all objects $$x$$ under consideration are in the same equivalence class with respect to the relation $$R$$
• “Prime factorization of an integer is unique up to ordering”
• This means that two lists of prime factors are equivalent because we can obtain one from the other by reordering the list
• With equivalence relations we can ignore subtleties
• “up to ordering” means “ignoring the particular ordering”
• Some other common examples include:
• “up to isomorphism”
• “up to permutations”
• “up to rotations”
• Informally, the word modulo is used for similar purposes, e.g. “modulo isomorphism”