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In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined , ill defined or ambiguous . [ 1 ]
The definition of a formula comes in several parts. First, the set of terms is defined recursively. Terms, informally, are expressions that represent objects from the domain of discourse. Any variable is a term. Any constant symbol from the signature is a term
It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. [note 1] [8] Some examples of mathematical statements that are computable include: All statements characterised in modern programming languages, including C++, Python, and ...
However, being well-formed is not enough to be considered well-defined. For example in arithmetic, the expression is well-formed, but it is not well-defined. (See Division by zero). Such expressions are called undefined.
For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an effective method, an algorithm can be expressed within a finite amount of space and time [3] and in a well-defined formal language [4] for calculating a function. [5]
Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element. The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds ...
Well-Ordered: The exact order of operations performed in an algorithm should be concretely defined. Feasibility: All steps of an algorithm should be possible (also known as effectively computable). Input: an algorithm should be able to accept a well-defined set of inputs.
Even for well-defined infinite expressions, the value of the infinite expression may be ambiguous or not well-defined; for instance, there are multiple summation rules available for assigning values to series, and the same series may have different values according to different summation rules if the series is not absolutely convergent.