MATH.7:4.2 - Establish the reversible map on its declared sets

Give h:X→Y and r:Y→X and establish both equations:

r(h(x))=x for every x in X;

h(r(y))=y for every y in Y.

They establish that r is the inverse of h. A formula, a complete finite table or an existing applicable result can supply the maps and these equations.

If h only covers part of the proposed Y, you can take its image h(X) as the receiving set when that answers the question. If h combines distinct source elements, returning the whole original element requires more information or a different map. A quotient may still retain the requested operation under MATH.2. MATH.6’s left-inverse examples show why recovery in one direction alone leaves the other direction open.

Keep restrictions in the sets. For example, squaring is reversible from nonnegative real numbers to nonnegative real numbers, with the nonnegative square root as inverse. Squaring on all real numbers combines opposite inputs and cannot support that inverse construction for the whole source.