Definition
Extensionality
Extensionality is the principle that objects are equal when they agree on their mathematical content, regardless of how they are described or constructed. For a type of object
, a family of observations is extensional if The observations must determine the object completely; agreement on an arbitrary selection of properties is not sufficient.
Two sets are equal exactly when they have the same elements:
Two functions
are equal exactly when their values agree at every input: This is function extensionality. For partial functions, equality also requires the same domain of definedness.
Example
Different expressions, the same function
Let
be given by and . Since for every , function extensionality gives . Their defining expressions differ, but their values agree at every input.