MATH.7:1 - Problem frame

Use this pattern when another representation makes a mathematical construction easier to work with and you need to carry its operations and results across a reversible map. You may want to calculate with coordinates, encode sets by bit vectors, or give a familiar carrier a different algebraic structure.

A bijection pairs every element of one set with one element of another and has an inverse in both directions. It can be used to define operations and relations on the receiving set. With those definitions, the map becomes an isomorphism of the chosen structures: it preserves their operations and reflects their relations.

Start with one needed operation and one input. Recover the input in the source, perform the source operation, and map its result back. Return the resulting operation and its domain, or the part of the proposed correspondence that cannot be reversed.

The reader needs functions, composition, inverse functions and elementary algebra. The main construction covers operations with finitely many inputs, constants and relations on sets. The worked coordinate example also uses squared Euclidean length. If the map and the needed preservation result are already available, use them. A one-way representation or a summary that deliberately forgets distinctions can instead use MATH.2, MATH.5 and FPF C.29.1 for the result it preserves.