C.29.1:4.2 - Construct the correspondence from the required distinctions

Choose a source domain X and a receiving domain Y. Construct a map F from the allowed source cases to their receiving representations. Explain its action in the working terms before relying on its notation: F(n,r) = n − r retains available stock and forgets the separate totals; a route endpoint summary forgets which route was taken.

A correspondence can use several maps. An operation may take one kind of input and return another, so its input map and output map can differ. For an operation with several inputs, say how each input is mapped and which combinations are permitted. Where the intended account relates one case to several possible representations, use that relation explicitly. Selecting one representation for each source case defines another construction; establish that the selection supports the intended conclusion.

Construct the correspondence by asking what the receiving question needs. Identify a candidate variable, relation or operation that carries that information. Compute its value on source cases. Determine which cases it combines. Try to express the receiving operation using only what remains. If the expression still depends on an omitted quantity, either retain that quantity or establish why it cancels for this use.

For a change of coordinates, construct the inverse when returning a complete source state is intended. For a summary, identify the source cases compatible with each receiving value. A many-to-one map can answer a particular question exactly even though it cannot reconstruct the whole source state.

Keep the domain visible. A division requires a nonzero denominator; a square-root substitution may impose a sign choice; a route may exist only for certain endpoints or histories. A map justified on one part of X supports conclusions on that part. A receiving value outside F(X), the set of represented source cases, needs a further argument before it is used as a source possibility.

The first result of this step is a usable correspondence with an explained meaning. It may be a formula, a small table, a diagram, or an already defined mathematical map. Its form follows the work.