“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”