Implementation of mathematics in set theory: Difference between revisions

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* A relation R is <strong>well-founded</strong> if for every set S which meets the field of R, there is x in S whose preimage under R does not meet S.

* A relation R is <strong>extensional</strong> if for every x,y in the field of R, x=y iff x and y have the same preimage under R.

Some kinds of relations:

* A relation R is an <strong>equivalence relation</strong> iff it is reflexive, symmetric, and transitive.

* A relation R is a <strong>partial order</strong> iff it is reflexive, antisymmetric, and transitive.

* A relation R is a <strong>linear order</strong> iff it is a partial order and for every x,y in the field of R, either <math>xRy</math> or <math>yRx</math>.

* A relation R is a <strong>well-ordering</strong> iff it is a linear order and it is well-founded.

* A relation R is a <strong>set picture</strong> iff it is well-founded and extensional.

== Functions ==