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In particular, the term well-defined is used with respect to (binary) operations on cosets. In this case, one can view the operation as a function of two variables, and the property of being well-defined is the same as that for a function. For example, addition on the integers modulo some n can be defined naturally in terms of integer addition.
Scholars distinguish between well-defined and ill-defined problems. Briggs and Reinig defined a well-defined solution in terms of space solution space. Pretz, Naples, and Sternberg defined a well-defined problem as one for which the parts of the solution are closely related or clearly based on the information given. Problem finding applies to ...
The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each.
Calculations or statements which are ill-defined, such that they cannot be unambiguously encoded into a Turing machine: ("Paul loves me twice as much as Joe"). Problem statements which do appear to be well-defined, but for which it can be proved that no Turing machine exists to solve them (such as the halting problem).
A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. In non-mathematical terms, an ill-conditioned problem is one where, for a small change in the inputs (the independent variables) there is a large change in the answer or dependent variable. This means ...
[14] [15] Design problems are typically wicked because they are often ill-defined (no prescribed way forward), involve stakeholders with different perspectives, and have no "right" or "optimal" solution. [16] Thus wicked problems cannot be solved by the application of standard (or known) methods; they demand creative solutions. [17] [18]
Hard systems approaches such as systems analysis (structured methods), operations research and so on, assume that the problems associated with such systems are well-defined and likely to have a single, optimum solution, so a problem-solving approach will work well as technical factors tend to predominate. [1] [2]
Problems that are not well-posed in the sense above are termed ill-posed. A simple example is a global optimization problem, because the location of the optima is generally not a continuous function of the parameters specifying the objective, even when the objective itself is a smooth function of those parameters.