Extensionality

From Academic Kids

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 functions f and g, we can say that they are equal if

f(x) = g(x)

for all x in the common function domain X. This extensional equality is the usual definition if the function range Y 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.

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