MATH.20:1 - Problem frame

Use this pattern when obtaining a mathematical value or object is difficult, but a justified comparison could already answer the receiving question. Construct something known to lie below, above, inside or outside the unknown in the relevant mathematical order.

Examples include bounding a best attainable value by a feasible construction and an inequality, enclosing a set between simpler sets, or converting an equation residual into a bound on solution error. The common method is to build and justify the comparison, then determine what it permits next.

First useful move: name the unknown and one comparison that would change the next action. A single feasible construction can settle an existence question or one side of an optimum bound. A proved enclosure can exclude a region. Construct only the bound the current question needs.

The reader needs elementary inequalities, sets and functions. The examples introduce their graph and set constructions; the residual example also uses linear equations and an inverse map. More specialized bounds require the properties of their particular mathematical objects.

If an available value or direct calculation answers the question more easily, use it. FPF’s characterization and choice methods select among alternatives; this pattern supplies mathematical bounds those choices may use. A mathematical comparison applied to a physical or organizational subject also requires the correspondence work in C.29 and MMP.