MATH.17:1 - Problem frame

Use this pattern when the rule for constructing or transforming something has itself become the object of work. You may need to combine allowable rules, apply a rule to a different kind of input, compare ways of repeating it, or change the rule while retaining a useful property. Knowing how to execute each individual rule leaves these questions open.

Start with the operation you want to perform on rules and the consequence you need from it. For example: “Can these permitted updates be composed?” or “Can I transform each stage separately and obtain the same result as transforming their composite?” One constructed operation, together with its applicability and the law or counterexample that answers that question, is a useful result.

Here a space of operations is a specified collection of operations with the equality, composition and further structure needed by the question. The main route uses sets and functions; it requires understanding function application, composition, sets and elementary equality arguments. MATH.16 explains functions as mathematical objects, evaluation and currying. The polynomial example is an optional branch using polynomial arithmetic.

Use an existing rule directly when its application settles the question. Construct a space of operations when you need to reason about, generate or change rules. A topology, metric or order on that space becomes part of the construction when the question needs continuity, approximation or comparison in that sense.