In mathematics, this usually refers to some form of the principle, going back to Leibniz, that two mathematical objects are equal if there is no test to distinguish them. For example, given two mathematical functionsf and g, we can say that they are equal if
f(x) = g(x)
for all x in the common function domainX. This extensional equality is the usual definition if the function rangeY is also common to the two functions. If, on the other hand, we distinguish functions by the data attached to them in the type theory sense, so that we could for example choose a larger set Z as range for one of them, that equality is not the same sense extensional. That is one sense in which extensionality may fail. Another one is that consideration of the process by which a function is computed, if taken into account, will usually contradict extensionality.
In axiomatic set theory, extensionality is expressed in the axiom of extension, which states that two sets are equal if and only if they contain the same elements. In lambda calculus, extensionality is expressed by the eta-conversion rule, which allows conversion between any two expressions that denote the same function.
Continuous extension may be described as that property in virtue whereof the parts into which material substance is divisible are situally arranged in orderly relation one beyond the other (internal and potentially local extension) and hence are naturally commensurate with the corresponding parts of the immediately environing surfaces (external and actual local extension).
Probably the more general opinion is that extension radically and essentially consists in the internal distribution of the parts into which matter is divisible, and that external extension, or the correspondence of those parts to the parts of the locating surfaces, is a sequent property of essential or internal extension.
Continuous extension is an objective property of matter, not a mere mental form moulding the sensuous impression produced in the sensory organs by some sort of physical motion.
A filenameextension is an extra set of (usually) alphanumeric Generally speaking, the term alphanumeric refers to anything that consists of only letters and numbers.
Filenameextensions have been in use for decades, but they have gained common usage because the file systems file system is a method for storing and organizing computer files and the data they contain to make it easy to find and access them.
Mapping filenameextensions to content-types is then done using different heuristics, such as examining both the filenameextension and the contents of the file.
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