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MyWikiBiz, Author Your Legacy — Saturday November 23, 2024
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→Projective reducibility of relations: cleanup
* Picture a ''k''-adic relation ''L'' as a body that resides in a ''k''-dimensional space ''X''. If the domains of the relation ''L'' are ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> , then the ''extension'' of the relation ''L'' is a subset of the cartesian product ''X'' = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> .
* Picture a ''k''-adic relation ''L'' as a body that resides in a ''k''-dimensional space ''X''. If the domains of the relation ''L'' are ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> , then the ''extension'' of the relation ''L'' is a subset of the cartesian product ''X'' = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> .
In this setting, the interval ''K'' = [1, ''k''] = {1, ;…, ''k''} is called the ''[[index set]]'' of the ''[[indexed family]]'' of sets ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> .
In this setting, the interval ''K'' = [1, ''k''] = {1, …, ''k''} is called the ''[[index set]]'' of the ''[[indexed family]]'' of sets ''X''<sub>1</sub>, …, ''X''<sub>''k''</sub> .
For any subset ''F'' of the index set ''K'', there is the corresponding subfamily of sets, {''X''<sub>''j''</sub> : ''j'' ∈ ''F'' }, and there is the corresponding cartesian product over this subfamily, notated and defined as ''X''<sub>''F''</sub> = <font size="+2">Π</font><sub>''j'' ∈ ''F''</sub> ''X''<sub>''j''</sub>.
For any subset ''F'' of the index set ''K'', there is the corresponding subfamily of sets, {''X''<sub>''j''</sub> : ''j'' ∈ ''F'' }, and there is the corresponding cartesian product over this subfamily, notated and defined as ''X''<sub>''F''</sub> = <font size="+2">Π</font><sub>''j'' ∈ ''F''</sub> ''X''<sub>''j''</sub>.
