View example sentences and word forms for Extensionality.
Extensionality
Extensionality meaning
The principle that objects are equal if and only if their observed properties are the same, regardless of internal processes that lead to those properties. | The principle that functions are equal if and only if they operate on the same domain and for any given element of the domain the result of each of the functions is the same. | The principle, codified in the axiom of extensionality, that sets are equal if and only if they contain the same elements.
Example sentences (7)
However, we say that two sets which differ only in that one has duplicate members are in fact exactly identical (see Axiom of extensionality ).
Interpretation We can use the axiom of extensionality to show that there is only one empty set.
Most of the axioms of equality still follow from the definition; the remaining one is : and it becomes this axiom that is referred to as the axiom of extensionality in this context.
See also * Extensionality for a general overview.
The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory.
The principle of extensionality in set theory assures us that any matching pair of curly braces enclosing one or more instances of the same individuals denote the same set.
While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.