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A '''relation''' <math>L\!</math> is defined by specifying two mathematical objects as its constituent parts:

A '''relation''' <math>L\!</math> is defined by specifying two mathematical objects as its constituent parts:

:* The first part is called the ''figure'' of <math>L,\!</math> notated as <math>\operatorname{figure}(L).</math> or <math>\operatorname{fig}(L).</math>

<p>The first part is called the ''figure'' of <math>L,\!</math> notated as <math>\operatorname{figure}(L).</math></p>

<p>The second part is called the ''ground'' of <math>L,\!</math> notated as <math>\operatorname{ground}(L).</math></p>

:* The second part is called the ''ground'' of <math>L,\!</math> notated as <math>\operatorname{ground}(L)</math> or <math>\operatorname{grd}(L).</math>

In the special case of a ''finitary relation'', for concreteness a ''<math>k\!</math>-place relation'', the concepts of figure and ground are defined as follows:

In the special case of a '''finitary relation''', for concreteness a '''<math>k\!</math>-place relation''', the concepts of figure and ground are defined as follows:

<p>The ''ground'' of <math>L\!</math> is a sequence of <math>k\!</math> nonempty sets, <math>X_1, \ldots, X_k,</math> called the ''domains'' of the relation <math>L.\!</math></p>

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<p>The ''figure'' of <math>L\!</math> is a subset of the cartesian product taken over the domains of <math>L.\!</math></p>

<p>In sum we have:  <math>\operatorname{figure}(L) ~\subseteq~ \operatorname{ground}(L) ~=~ X_1 \times \ldots, \times X_k.</math></p>

:* The '''ground''' of ''L'' is a [[sequence]] of ''k'' [[nonempty]] [[set]]s, ''X''₁,&nbsp;…,&nbsp;''X''_''k'', called the ''domains'' of the relation ''L''.

Strictly speaking, then, the relation <math>L\!</math> is an ordered pair of mathematical objects, <math>L = (\operatorname{figure}(L), \operatorname{ground}(L)),</math> but it is customary in loose speech to use the single name <math>L\!</math> in a systematically equivocal fashion, taking it to denote either the pair <math>L = (\operatorname{figure}(L), \operatorname{ground}(L))</math> or the figure <math>\operatorname{figure}(L).</math> There is usually no confusion about this so long as the ground of the relation can be gathered from context.

 

:* The '''figure''' of ''L'' is a [[subset]] of the [[cartesian product]] taken over the domains of ''L'', that is, ''F''(''L'')&nbsp;&sube;&nbsp;''G''(''L'') = ''X''₁&nbsp;&times;&nbsp;…&nbsp;&times;&nbsp;''X''_''k''.

 

Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''),&nbsp;''G''(''L'')), but it is customary in loose speech to use the single name ''L'' in a systematically equivocal fashion, taking it to denote either the couple ''L'' = (''F''(''L''),&nbsp;''G''(''L'')) or the figure ''F''(''L'').  There is usually no confusion about this so long as the ground of the relation can be gathered from context.

==Definition==

==Definition==