Difference between revisions of "Grounded relation"
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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. | ||
| − | 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''. |
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. | ||
Revision as of 05:14, 9 January 2008
A grounded relation over a sequence of sets 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 domains 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, X1, …, Xk, then L has the form L = (F(L), G(L)), where F(L) ⊆ G(L) = X1 × … × Xk, 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
- Relation composition
- Relation construction
- Relation reduction
- Relation theory
- 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.
