Implementation of mathematics in set theory: Difference between revisions

Revision as of 18:38, 18 January 2013

Revision as of 18:40, 18 January 2013

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=== Properties and kinds of relations ===

Let ''R'' be some [[binary relation]]. ''R'' is:

*'''[[Reflexive relation|Reflexive]]''' if ''xRx'' for every ''x'' in the ___domain or rangefield of ''R''.

* '''[[Symmetric relation|Symmetric]]''' if <math>\forall x,y \,(xRy \to yRx)</math>.

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Relations having certain combinations of the above properties have standard names. ''R'' is:

* An '''[[equivalence relation]]''' [[if and only if]] ''R'' is reflexive, symmetric, and transitive.

* A '''[[partial order]]''' if and only if ''R'' is reflexive, antisymmetric, and transitive.

* A '''[[linear order]]''' if and only if ''R'' is a partial order and for every ''x,y'' in the field of ''R'', either ''xRy'' or ''yRx''.

* A '''[[well-ordering]]''' if and only if ''R'' is a linear order and well-founded.

* A '''set picture''' if and only if ''R'' is well-founded and extensional, and the field of ''R'' either equals the downward closure of one of its members (called its ''top element''), or is empty.

== Functions ==