# “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”

## References