Gallium trichloride: Difference between revisions
en>Firilacroco +ro |
en>Addbot m Bot: Migrating 7 interwiki links, now provided by Wikidata on d:q31153 (Report Errors) |
||
| Line 1: | Line 1: | ||
In [[mathematics]], a [[partial order]] < on a set ''X'' is said to be '''dense''' if, for all ''x'' and ''y'' in ''X'' for which ''x'' < ''y'', there is a ''z'' in ''X'' such that ''x'' < ''z'' < ''y''. | |||
The [[rational number]]s with the ordinary ordering are a densely ordered set in this sense, as are the [[real number]]s. On the other hand, the ordinary ordering on the [[integer]]s is not dense. | |||
==Generalizations== | |||
Any [[binary relation]] ''R'' is said to be ''dense'' if, for all ''R''-related ''x'' and ''y'', there is a ''z'' such that ''x'' and ''z'' and also ''z'' and ''y'' are ''R''-related.{{Citation needed|reason=There are reasonable, inequivalent alternatives, so a source should be given.|date=March 2012}} Formally: | |||
: <math> \forall x\ \forall y\ xRy\Rightarrow (\exists z\ xRz \land zRy).</math> | |||
Every [[reflexive relation]] is dense. A strict [[partial order]] < is a dense order [[iff]] < is a dense relation. | |||
==See also== | |||
*[[Dense set]] | |||
*[[Dense-in-itself]] | |||
*[[Kripke semantics]] | |||
== References == | |||
* [[David Harel]], [[Dexter Kozen]], Jerzy Tiuryn, ''Dynamic logic'', MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff | |||
[[Category:Mathematical relations]] | |||
[[Category:Order theory]] | |||
{{mathlogic-stub}} | |||
Revision as of 12:03, 26 February 2013
In mathematics, a partial order < on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y.
The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers. On the other hand, the ordinary ordering on the integers is not dense.
Generalizations
Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Formally:
Every reflexive relation is dense. A strict partial order < is a dense order iff < is a dense relation.
See also
References
- David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0-262-08289-6, p. 6ff