| 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. |