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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. | | 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. |
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− | For example, if ''L'' is a grounded relation over a finite sequence of sets, then ''L'' has the form ''L'' = (''F''(''L''), ''G''(''L'')), where ''F''(''L'') ⊆ ''G''(''L'') = ''X''<sub>1</sub> × … × ''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>, …, ''X''<sub>''k''</sub> , then ''L'' has the form ''L'' = (''F''(''L''), ''G''(''L'')), where ''F''(''L'') ⊆ ''G''(''L'') = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> , for some positive integer ''k''. |
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| 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. | | 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. |
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| * [[Relation theory]] | | * [[Relation theory]] |
| * [[Relation type]] | | * [[Relation type]] |
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− | ==Document history==
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− | 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.
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− | * [http://en.wikipedia.org/wiki/Grounded_relation Grounded relation], [http://en.wikipedia.org/ Wikipedia].
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| [[Category:Logic]] | | [[Category:Logic]] |
| [[Category:Mathematics]] | | [[Category:Mathematics]] |
− | [[Category:Set theory]] | + | [[Category:Set Theory]] |