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MyWikiBiz, Author Your Legacy — Wednesday May 01, 2024
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The formal definition of a '''finite arity relation''', specifically, a '''k-ary relation''' can now be stated.
The formal definition of a '''finite arity relation''', specifically, a '''k-ary relation''' can now be stated.
* '''Definition.''' A '''k-ary relation''' ''L'' over the nonempty sets ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> is a (1+''k'')-tuple ''L'' = (''F''(''L''), ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>) where ''F''(''L'') is a subset of the cartesian product ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub>. If all of the ''X''<sub>''j''</sub> for ''j'' = 1 to ''k'' are the same set ''X'', then ''L'' is more simply called a '''''k''-ary relation over ''X'''''. The set ''F''(''L'') is called the ''figure'' of ''L'' and, providing that the sequence of sets ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> is fixed throughout a given discussion or determinate in context, one may regard the relation ''L'' as being determined by its figure ''F''(''L'').
* '''Definition.''' A '''k-ary relation''' ''L'' over the nonempty sets ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> is a (1+''k'')-tuple ''L'' = (''F''(''L''), ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>) where ''F''(''L'') is a subset of the cartesian product ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub>. If all of the ''X''<sub>''j''</sub> for ''j'' = 1 to ''k'' are the same set ''X'', then ''L'' is more simply called a '''''k''-ary relation over ''X'''''. The set ''F''(''L'') is called the ''figure'' of ''L'' and, provided that the sequence of sets ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> is fixed throughout a given discussion or is otherwise determinate in context, one may regard the relation ''L'' as being determined by its figure ''F''(''L'').
The formal definition simply repeats more concisely what was said above, merely unwrapping the conceptual packaging of the relation's ground to define the relation in 1 + ''k'' parts, as ''L'' = (''F''(''X''), ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>), rather than just the two, as ''L'' = (''F''(''L''), ''G''(''L'')).
The formal definition simply repeats more concisely what was said above, merely unwrapping the conceptual packaging of the relation's ground to define the relation in 1 + ''k'' parts, as ''L'' = (''F''(''X''), ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>), rather than just the two, as ''L'' = (''F''(''L''), ''G''(''L'')).
A ''k''-ary '''predicate''' is a ''[[boolean-valued function]]'' on ''k'' variables.
A ''k''-ary '''predicate''' is a ''[[boolean-valued function]]'' on ''k'' variables.
