Concise Guide to Formal Methods by Gerard O'Regan

Author:Gerard O'Regan
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

7.5 Undefined Values

Total functions f : X → Y are functions that are defined for every element in their domain, and total functions are widely used in mathematics. However, there are functions that are undefined for one or more elements in their domain, and one example is the function y = 1/ x which is undefined at x = 0.

Partial functions arise naturally in computer science, and such functions may fail to be defined for one or more values in their domain. One approach to dealing with partial functions is to employ a precondition, which restricts the application of the function to where it is defined. This makes it possible to define a new set (a proper subset of the domain of the function) for which the function is total over the new set.

Undefined terms often arise1 and need to be dealt with. Consider, the example of the square root function √x taken from [5]. The domain of this function is the positive real numbers, and the following expression is undefined:

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