Grounded relation - Revision history

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A '''grounded relation''' over a [[sequence]] of [[set]]s is a mathematical object consisting of two components. The first component is a subset of the [[cartesian product]] taken over the given sequence of sets, which sets are called the ''[[domain of discourse|domain]]s'' of the relation. The second component is just the cartesian product itself.

For example, if ''L'' is an grounded relation over a finite sequence of sets, then ''L'' has the form ''L'' = (''F''(''L''), ''G''(''L'')), where ''F''(''L'') &sube; ''G''(''L'') = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub>, for some positive integer ''k''.

The default assumption in almost all applied settings is that the domains of the grounded relation are [[nonempty]] sets, hence departures from this assumption need to be noted explicitly.

==See also==

* [[Logic of relatives]]
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
* [[Relation construction]]
* [[Relation reduction]]
* [[Relation theory]]
* [[Relation type]]

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 [[Category:Logic]]
 [[Category:Mathematics]]
-[[Category:Set theory]]
+[[Category:Set Theory]]

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 * [[Relation theory]]
 * [[Relation type]]
 [[Category:Logic]]
 [[Category:Mathematics]]
 [[Category:Set theory]]

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 A '''grounded relation''' over a [[sequence]] of [[set]]s is a mathematical object consisting of two components.  The first component is a subset of the [[cartesian product]] taken over the given sequence of sets, which sets are called the ''[[domain of discourse|domain]]s'' of the relation.  The second component is just the cartesian product itself.
-For example, if ''L'' is a grounded relation over a finite sequence of sets, ''X''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''X''<sub>''k''</sub>, then ''L'' has the form ''L''&nbsp;=&nbsp;(''F''(''L''),&nbsp;''G''(''L'')), where ''F''(''L'')&nbsp;&sube;&nbsp;''G''(''L'') = ''X''<sub>1</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''<sub>''k''</sub>, for some positive integer ''k''.
+For example, if ''L'' is a grounded relation over a finite sequence of sets, ''X''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''X''<sub>''k''</sub>&nbsp;, then ''L'' has the form ''L''&nbsp;=&nbsp;(''F''(''L''),&nbsp;''G''(''L'')), where ''F''(''L'')&nbsp;&sube;&nbsp;''G''(''L'') = ''X''<sub>1</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''<sub>''k''</sub>&nbsp;, for some positive integer ''k''.
 The default assumption in almost all applied settings is that the domains of the grounded relation are [[nonempty]] sets, hence departures from this assumption need to be noted explicitly.

@@ Line 1: / Line 1: @@
 A '''grounded relation''' over a [[sequence]] of [[set]]s is a mathematical object consisting of two components.  The first component is a subset of the [[cartesian product]] taken over the given sequence of sets, which sets are called the ''[[domain of discourse|domain]]s'' of the relation.  The second component is just the cartesian product itself.
-For example, if ''L'' is a grounded relation over a finite sequence of sets, then ''L'' has the form ''L''&nbsp;=&nbsp;(''F''(''L''),&nbsp;''G''(''L'')), where ''F''(''L'')&nbsp;&sube;&nbsp;''G''(''L'') = ''X''<sub>1</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''<sub>''k''</sub>, for some positive integer ''k''.
+For example, if ''L'' is a grounded relation over a finite sequence of sets, ''X''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''X''<sub>''k''</sub>, then ''L'' has the form ''L''&nbsp;=&nbsp;(''F''(''L''),&nbsp;''G''(''L'')), where ''F''(''L'')&nbsp;&sube;&nbsp;''G''(''L'') = ''X''<sub>1</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''<sub>''k''</sub>, for some positive integer ''k''.
 The default assumption in almost all applied settings is that the domains of the grounded relation are [[nonempty]] sets, hence departures from this assumption need to be noted explicitly.

@@ Line 1: / Line 1: @@
 A '''grounded relation''' over a [[sequence]] of [[set]]s is a mathematical object consisting of two components.  The first component is a subset of the [[cartesian product]] taken over the given sequence of sets, which sets are called the ''[[domain of discourse|domain]]s'' of the relation.  The second component is just the cartesian product itself.
-For example, if ''L'' is an grounded relation over a finite sequence of sets, then ''L'' has the form ''L'' = (''F''(''L''), ''G''(''L'')), where ''F''(''L'') &sube; ''G''(''L'') = ''X''<sub>1</sub> &times; &hellip; &times; ''X''<sub>''k''</sub>, for some positive integer ''k''.
+For example, if ''L'' is a grounded relation over a finite sequence of sets, then ''L'' has the form ''L''&nbsp;=&nbsp;(''F''(''L''),&nbsp;''G''(''L'')), where ''F''(''L'')&nbsp;&sube;&nbsp;''G''(''L'') = ''X''<sub>1</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;''X''<sub>''k''</sub>, for some positive integer ''k''.
 The default assumption in almost all applied settings is that the domains of the grounded relation are [[nonempty]] sets, hence departures from this assumption need to be noted explicitly.

← Older revision		Revision as of 03:06, 9 January 2008
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	* [[Relation theory]]		* [[Relation theory]]
	* [[Relation type]]		* [[Relation type]]
		+
		+	==Document history==
		+
		+	Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
		+
		+	* [http://en.wikipedia.org/wiki/Grounded_relation Grounded relation], [http://en.wikipedia.org/ Wikipedia].
		+
		+	[[Category:Logic]]
		+	[[Category:Mathematics]]
		+	[[Category:Set theory]]