MATH.7:4 - Solution
Choose the structure to carry → establish both inverse equations → transport operations and relations → derive the retained laws → calculate and return the result.
MATH.7:4.1 - Choose the source structure and the receiving use
Name the source set X and the operations, constants or relations needed by the question. For an additive structure these might be addition, zero and negation. For an ordered structure the comparison relation matters too. Include a quantity such as length when the requested conclusion depends on it.
Name the receiving set Y and distinguish two tasks: constructing its operations, or showing that operations already required there correspond to the source. In the second task, retain those required operations for the comparison. Replacing them with different transported operations changes the proposed mathematical structure.
For a single update, C.29.1’s correspondence comparison can be sufficient. Use the construction below when you need the related operations, their laws, a reusable representation or a way to move solutions between the two structures.
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.
MATH.7:4.3 - Define the receiving operations and relations
For a source operation op_X:X^n→X, define:
op_Y(y1,...,yn)=h(op_X(r(y1),...,r(yn))).
In words: decode each input, perform the source operation, then encode the output. Constants have no inputs, so a source constant c becomes h(c). A source relation R becomes:
R_Y(y1,...,yn) holds exactly when R_X(r(y1),...,r(yn)) holds.
The construction can have different input and output sets. For op:X1×...×Xn→Z, use a bijection h_i:Xi→Yi for each input and k:Z→W for the output. Then:
op_target(y1,...,yn)=k(op(h_1^-1(y1),...,h_n^-1(yn))).
An unchanged scalar result uses the identity map on that scalar set. Thus transporting a length calculation changes its input coordinates while retaining the numerical length.
For a partial source operation, transport its domain too. The receiving tuple is allowed precisely when its decoded tuple is in the source domain; define its output there by the same formula. A larger independently supplied receiving domain needs its own comparison before its extra inputs are used.
Calculate one small case in both descriptions. Use it to check the direction of the maps, the constants and the operation being performed. The general preservation result comes from the defining formula and inverse equations.
MATH.7:4.4 - Derive the laws that the chosen structure retains
For each operation, substituting h(x_i) into its definition and cancelling r(h(x_i)) gives:
op_Y(h(x1),...,h(xn))=h(op_X(x1,...,xn)).
Any other receiving operation with this equality must coincide with the constructed one: every receiving input has the form h(x_i). Thus the source operation and h determine the transported operation uniquely on Y.
To carry an equation built from these operations, follow its expressions from the variables and constants through each operation. Variables are mapped by h, constants by their specified images, and the displayed equality carries each composite expression. This is an induction on the expression’s construction; MATH.4 supplies that form of argument.
For expressions s and t in one carrier, it yields:
s_Y(h(x1),...)=h(s_X(x1,...)) and t_Y(h(x1),...)=h(t_X(x1,...)).
An equality of the source expressions therefore gives equality of the receiving expressions. Conversely, h is injective, so equality of those receiving values gives equality of the source values. Surjectivity covers all assignments in Y. The same reasoning with the map for each sort treats operations with different input and output sets.
This transfers equational laws such as associativity, an identity law or distributivity for the operations actually carried. A further order, distance or other relation must use its own transported definition or an established compatibility result. For partial operations, compare the definedness of the compound expressions as well as their values.
MATH.7:4.5 - Calculate in the useful representation and recover the answer
Translate the inputs, constants and requested relation. Calculate or solve the corresponding problem using the receiving operations. Map the result back with the inverse appropriate to its kind.
For example, a source equation t_X(x)=b becomes t_Y(y)=h(b), where y=h(x) and all other parameters in t are translated too. A receiving solution y returns x=r(y). The correspondence gives both directions, so it also carries absence or uniqueness of a solution when the equations range over the declared sets.
Use a simpler formula for a transported operation after establishing equality with its defining formula. This can avoid repeated encoding and decoding. Compare the resulting calculation effort with direct source calculation when the representation is being chosen for efficiency. Preservation alone establishes no speed advantage.
If Y already had a required operation, compare it with the constructed one before transferring its laws or solutions. A disagreement can lead to a changed map, a different receiving structure, or continued use of the original description.
MATH.7:4.6 - Revisit only what the changed question uses
When h changes, derive the affected receiving operations and constants again. When the question adds a quantity or relation, construct its receiving form even if the earlier operations are unchanged. When a domain restriction changes, revisit both inverse equations and the allowed operation inputs.
Return a failed equation or two source cases that the map cannot distinguish through MATH.6 or MATH.2 as appropriate. Use C.29 when the mathematical representation is being related to another subject. Stop with the needed calculation, reusable transported structure, or identified obstruction.