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[[Table of mathematical symbols|'''Table of Mathematical Symbols''']]
==New Biz==
===Typographical Towers===
'''Version 1'''
'''Example 1'''. Any algebra being trivially a homologue of itself, the algebra of finitary operations on {0, 1} qualifies as a Boolean algebra. To understand the operations of Boolean algebra and their laws in general it therefore suffices to understand them for just this two-element Boolean algebra.
There being ''k''<sup>''k''</sup><sup><sup>''n''</sup></sup> ''n''-ary operations ''f'': ''X''<sup>''n''</sup>→''X'' on a ''k''-element set ''X'', there are therefore 2<sup>2</sup><sup><sup>''n''</sup></sup> ''n''-ary operations on {0,1}. Although we don't need to specify an order for the operations, it is natural to list the smaller arities first. This then makes the signature of a Boolean algebra
: 0-0-1-1-1-1-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-3-3-3-…,
meaning that every Boolean algebra, however small or large, has two constants or "nullary" operations, four unary operations, 16 binary operations, 256 ternary, etc., which we call the '''[[Boolean operation]]s''' of the given Boolean algebra.
'''Version 2'''
'''Example 1'''. Any algebra being trivially a homologue of itself, the algebra of finitary operations on {0, 1} qualifies as a Boolean algebra. To understand the operations of Boolean algebra and their laws in general it therefore suffices to understand them for just this two-element Boolean algebra.
There being ''k''<sup>''k''</sup><sup><sup>''n''</sup></sup> ''n''-ary operations ''f'': ''X''<sup>''n''</sup>→''X'' on a ''k''-element set ''X'', there are therefore 2<sup>2</sup><sup><sup>''n''</sup></sup> ''n''-ary operations on {0,1}. Although we don't need to specify an order for the operations, it is natural to list the smaller arities first. This then makes the signature of a Boolean algebra
: 0-0-1-1-1-1-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-3-3-3-…,
meaning that every Boolean algebra, however small or large, has two constants or "nullary" operations, four unary operations, 16 binary operations, 256 ternary, etc., which we call the '''[[Boolean operation]]s''' of the given Boolean algebra.
===Themes and variations===
{| class="wikitable" align="center" style="text-align:center; width:90%"
|-
! Laws \ Explananda !! Particular Facts !! General Regularities
|-
! Universal Laws
| D-N <p> Deductive-Nomological || D-N <p> Deductive-Nomological
|-
! Statistical Laws
| I-S <p> Inductive-Statistical || D-S <p> Deductive-Statistical
|}
{| class="wikitable"
|+Multiplication table
|-
! × !! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|-
! 4
| 4 || 8 || 12
|-
! 5
| 5 || 10 || 15
|}
{| class="wikitable"
|+Multiplication table
|-
! × !! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|-
! 4
| 4 || 8 || 12
|-
! 5
| 5 || 10 || 15
|}
===Truth and its vicissitudes===
====Vicissitude 1====
'''''Truth''''' ([[antonym]] ''[[false|falsity]]'') refers to the property of a [[proposition]] or its [[symbol]]ic expression having a degree of fidelity with ''[[reality]]''. A statement that is judged to have the property of truth is said to be ''true'', and may be referred to in substantive terms as "a truth". The [[hypostatic object|abstract object]] to which all true statements may be taken to refer is frequently referred to in general terms as "the truth". In [[rhetoric]]al contexts where [[obfuscation]] is a factor, ''[[honesty]]'' and ''[[sincerity]]'' may also be considered as aspects of the "truth".
====Vicissitude 2====
'''''Truth''''' ([[antonym|opposite]] ''[[false|falsity]]'') refers to the property of a [[proposition]] or its [[symbol]]ic expression as having a strong fidelity with ''[[reality]]''. A statement that is judged to have the property of truth is said to be ''true'', and may be referred to in substantive terms as "a truth". The [[hypostatic object|abstract object]] to which all true statements may be taken to refer is also referred to in general terms as "the truth". In [[rhetoric]]al contexts where [[obfuscation]] is a factor, ''[[honesty]]'' and ''[[sincerity]]'' may also be considered as aspects of the "truth".
==Old Biz==
{| cellpadding="2"
|-
| style="width:20px" |
| align="center" | <math>(G \circ H)_{ij}</math>
|
|-
|
| align="center" | <math>=\!</math>
| the <math>ij</math><sup>th</sup> entry in the matrix representation for <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the entry in the <math>i</math><sup>th</sup> row and the <math>j</math><sup>th</sup> column of <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the scalar product of the <math>i</math><sup>th</sup> row of <math>G\!</math> with the <math>j</math><sup>th</sup> column of <math>H\!</math>
|-
|
| align="center" | <math>=\!</math>
| <math>\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}</math>
|}
{| border="1" cellpadding="4"
|-
| style="width:20px" |
| align="center" | <math>(G \circ H)_{ij}</math>
|
|-
|
| align="center" | <math>=\!</math>
| the <math>ij</math><sup>th</sup> entry in the matrix representation for <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the entry in the <math>i</math><sup>th</sup> row and the <math>j</math><sup>th</sup> column of <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the scalar product of the <math>i</math><sup>th</sup> row of <math>G\!</math> with the <math>j</math><sup>th</sup> column of <math>H\!</math>
|-
|
| align="center" | <math>=\!</math>
| <math>\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}</math>
|}
{| border="1" cellspacing="4"
|-
| style="width:20px" |
| align="center" | <math>(G \circ H)_{ij}</math>
|
|-
|
| align="center" | <math>=\!</math>
| the <math>ij</math><sup>th</sup> entry in the matrix representation for <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the entry in the <math>i</math><sup>th</sup> row and the <math>j</math><sup>th</sup> column of <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the scalar product of the <math>i</math><sup>th</sup> row of <math>G\!</math> with the <math>j</math><sup>th</sup> column of <math>H\!</math>
|-
|
| align="center" | <math>=\!</math>
| <math>\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}</math>
|}
The formula for computing ''G'' o ''H'' says the following:
(G o H)_ij
= the ij^th entry in the matrix representation for G o H
= the entry in the i^th row and the j^th column of G o H
= the scalar product of the i^th row of G with the j^th column of H
= Sum_k (G_ik H_kj)
{|
|-
| align="center" | <math>(G \circ H)_{ij}</math>
|
|
|-
|
| <math>=\;</math>
| the ij^th entry in the matrix representation for G o H
|-
|
| <math>=\;</math>
| the entry in the i^th row and the j^th column of G o H
|-
|
| <math>=\;</math>
| the scalar product of the i^th row of G with the j^th column of H
|-
|
| <math>=\;</math>
| Sum_k (G_ik H_kj)
|}
: <math>(G \circ H)_{ij}</math>
{|
|-
| style="width:20px" |
| =
| the ij^th entry in the matrix representation for G o H
|-
|
| =
| the entry in the i^th row and the j^th column of G o H
|-
|
| =
| the scalar product of the i^th row of G with the j^th column of H
|-
|
| =
| <math>\sum_{k} (G_{ik} H_{kj})</math>
|}
==Matrix Matters==
'''Table Format'''
{|
|-
| style="width:20px" |
| <math>F\ </math>
| <math>=\ 4:3:4</math>
| <math>+\ 4:4:4</math>
| <math>+\ 4:5:4</math>
|-
|
| <math>G\ </math>
| <math>=\ 4:3</math>
| <math>+\ 4:4</math>
| <math>+\ 4:5</math>
|-
|
| <math>H\ </math>
| <math>=\qquad\!\! 3:4</math>
| <math>+\qquad\!\! 4:4</math>
| <math>+\qquad\!\! 5:4</math>
|}
'''Matrix Format'''
: <math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\
G & = & 4:3 & + & 4:4 & + & 4:5 \\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
==Minimal Negation Operators==
:: ( ) = 0
:: (''x'') = ~''x'' = ¬''x'' = ''x''′
:: (''x'', ''y'') = ''x'' + ''y'' = ''x''′''y'' + ''xy''′
:: (''x'', ''y'', ''z'') = ''x''′''yz'' + ''xy''′''z'' + ''xyz''′
: <math>\begin{matrix}
(\ ) & = & 0 & = & \mbox{false} \\
(x) & = & \neg x & = & \tilde{x} & = & x' \\
(x, y) & = & x + y & = & \tilde{x} y \lor x \tilde{y} & = & x'y \lor xy'
\end{matrix}</math>
==Nested Tables==
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
{| border=1
| α
| align="center" | cell2
{| border=2 style="background-color:#ABCDEF;"
| NESTED
|-
| TABLE
|}
| valign="bottom" | the original table again
|}
{| border=1
| α
| align="center" | cell2
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_0 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 0\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
| valign="bottom" | the original table again
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_0 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 0\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_1 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 1\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
==Relational Data==
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f9f9f9; font-weight:bold; text-align:center; width:90%"
|+ '''Table 1. Relational Database'''
|- style="background:#efefef"
! style="width:15%" | Domain 1
! style="width:15%" | Domain 2
! style="width:15%" | ...
! style="width:15%" | Domain j
! style="width:15%" | ...
! style="width:15%" | Domain k
|-
| x<sub>11</sub>
| x<sub>12</sub>
| ...
| x<sub>1j</sub>
| ...
| x<sub>1k</sub>
|-
| x<sub>21</sub>
| x<sub>22</sub>
| ...
| x<sub>2j</sub>
| ...
| x<sub>2k</sub>
|-
| ... || ... || ... || ... || ... || ...
|-
| x<sub>i1</sub>
| x<sub>i2</sub>
| ...
| x<sub>ij</sub>
| ...
| x<sub>ik</sub>
|-
| ... || ... || ... || ... || ... || ...
|-
| x<sub>m1</sub>
| x<sub>m2</sub>
| ...
| x<sub>mj</sub>
| ...
| x<sub>mk</sub>
|}
==Dyadic Projections==
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
==Projective reducibility of triadic relations==
{{main|Triadic relation}}
By way of illustrating the different sorts of things that can occur in considering the projective reducibility of relations, it is convenient to reuse the four examples of 3-adic relations that are discussed in the main article on that subject.
===Examples of projectively irreducible relations===
The 3-adic relations '''L'''<sub>0</sub> and '''L'''<sub>1</sub> are shown in the next two Tables:
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_0 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 0\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_1 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 1\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
A ''2-adic projection'' of a 3-adic relation ''L'' is the 2-adic relation that results from deleting one column of the table for ''L'' and then deleting all but one row of any resulting rows that happen to be identical in content. In other words, the multiplicity of any repeated row is ignored.
In the case of the above two relations, '''L'''<sub>0</sub>, '''L'''<sub>1</sub> ⊆ ''X'' × ''Y'' × ''Z'' <u>≈</u> '''B'''<sup>3</sup>, the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
It is clear by inspection that the following equations hold:
{| align="center" cellpadding="4" style="width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|}
These equations say that '''L'''<sub>0</sub> and '''L'''<sub>1</sub> cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, either relation is said to be ''irreducible with respect to 2-adic projections''. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is ''projectively irreducible'', the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations.
===Examples of projectively reducible relations===
The 3-adic relations '''L'''<sub>A</sub> and '''L'''<sub>B</sub> are shown in the next two Tables:
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_A = \mbox{Sign Relation of Interpreter A}</math>
|-
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_B = \mbox{Sign Relation of Interpreter B}</math>
|-
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
==Epigraphs==
===Epigraph 1===
{|
| style="width:44%" |
| ''All rising to Great Place is by a Winding Staire''
|-
| style="width:44%" |
| Francis Bacon, ''Essays, Civil and Moral'' (1625)
|}
===Epigraph 2===
{|
| style="width:44%" |
| ''Hit's a-comin', boys. Tell yore folks hit's a-comin'.''
|-
| style="width:44%" |
| Thomas Wolfe, ''O Lost, A Story of the Buried Life''
|}
===Epigraph 3===
{|
||
| Ye knowe eek, that in forme of speche is chaunge
|-
||
| With-inne a thousand yeer, and wordes tho
|-
||
| That hadden prys, now wonder nyce and straunge
|-
||
| Us thinketh hem; and yet they spake hem so,
|-
||
| And spedde as wel in love as men now do;
|-
||
| Eek for to winne love in sondry ages,
|-
||
| In sondry londes, sondry been usages.
|-
| style="width:40%" |
| [[Geoffrey Chaucer]], [http://en.wikisource.org/wiki/Troilus_and_Criseyde ''Troilus and Criseyde'' (1385)]
|}
Ye knowe eek, that in forme of speche is chaunge
With-inne a thousand yeer, and wordes tho
That hadden prys, now wonder nyce and straunge
Us thinketh hem; and yet they spake hem so,
And spedde as wel in love as men now do;
Eek for to winne love in sondry ages,
In sondry londes, sondry been usages.
Geoffrey Chaucer, "Troilus and Criseyde", 2.4.22-28 (1385)
http://en.wikisource.org/wiki/Troilus_and_Criseyde:Book_II
===Epigraph 4===
{|
| style="width:44%" |
| Men loven of propre kinde newfangelnesse,
|-
| style="width:44%" |
| As briddes doon that men in cages fede.
|-
| style="width:44%" |
| — Geoffrey Chaucer, "The Squire's Tale"
|}
{|
| style="width:44%" |
| Whan it cam him to purpos for to reste,
|-
|
| I trowe he hadde thilke text in minde,
|-
|
| That 'alle thing, repeiring to his kinde,
|-
|
| Gladeth him-self'; thus seyn men, as I gesse;
|-
|
| Men loven of propre kinde newfangelnesse,
|-
|
| As briddes doon that men in cages fede.
|-
| style="width:44%" |
| — Geoffrey Chaucer, "The Squire's Tale"
|}
==Laws of Form==
===Formal Axioms===
'''Format 1'''
Here is one way of reading the axioms under the entitative interpretation:
:* I<sub>1</sub>. true or true = true.
:* I<sub>2</sub>. not true = false.
:* J<sub>1</sub>. ''a'' or not ''a'' = true.
:* J<sub>2</sub>. [''a'' or ''b''] and [''a'' or ''c''] = ''a'' or [''b'' and ''c''].
Here is one way of reading the axioms under the existential interpretation:
:* I<sub>1</sub>. false and false = false.
:* I<sub>2</sub>. not false = true.
:* J<sub>1</sub>. ''a'' and not ''a'' = false.
:* J<sub>2</sub>. [''a'' and ''b''] or [''a'' and ''c''] = ''a'' and [''b'' or ''c''].
'''Format 2'''
Here is one way of reading the axioms under the entitative interpretation:
{| align="center" cellpadding="4" style="width:80%"
| style="width:10%" | I<sub>1</sub>
| style="width:30%" | true or true
| style="width:10%" | =
| style="width:30%" | true
|-
| I<sub>2</sub>
| not true
| =
| false
|-
| J<sub>1</sub>
| ''a'' or not ''a''
| =
| true
|-
| J<sub>2</sub>.
| [''a'' or ''b''] and [''a'' or ''c'']
| =
| ''a'' or [''b'' and ''c'']
|}
Here is one way of reading the axioms under the existential interpretation:
{| align="center" cellpadding="4" style="width:80%"
| style="width:10%" | I<sub>1</sub>
| style="width:30%" | false and false
| style="width:10%" | =
| style="width:30%" | false
|-
| I<sub>2</sub>
| not false
| =
| true
|-
| J<sub>1</sub>
| ''a'' and not ''a''
| =
| false
|-
| J<sub>2</sub>
| [''a'' and ''b''] or [''a'' and ''c'']
| =
| ''a'' and [''b'' or ''c'']
|}
'''Format 3'''
{{col-begin}}
{{col-2}}
* A
* B
* C
* D
{{col-2}}
* 1
* 2
* 3
* 4
{{col-end}}
===Peirce's Law===
'''Format 1'''
Here is Peirce's own statement of the law:
<blockquote>
A ''fifth icon'' is required for the principle of [[excluded middle]] and other propositions connected with it. One of the simplest formulae of this kind is:
</blockquote>
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
<blockquote>
This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent ''x'' being false while its antecedent (''x'' —< ''y'') —< ''x'' is true. If this is true, either its consequent, ''x'', is true, when the the whole formula would be true, or its antecedent ''x'' —< ''y'' is false. But in the last case the antecedent of ''x'' —< ''y'', that is ''x'', must be true. (Peirce, CP 3.384).
</blockquote>
Peirce goes on to point out an immediate application of the law:
<blockquote>
From the formula just given, we at once get:
</blockquote>
{| align="center"
| {(''x'' —< ''y'') —< ''a''} —< ''x'',
|}
<blockquote>
where the ''a'' is used in such a sense that (''x'' —< ''y'') —< ''a'' means that from (''x'' —< ''y'') every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of ''x'' follows the truth of ''x''. (Peirce, CP 3.384).
</blockquote>
'''Format 2'''
Here is Peirce's own statement of the law:
: A ''fifth icon'' is required for the principle of [[excluded middle]] and other propositions connected with it. One of the simplest formulae of this kind is:
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
: This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent ''x'' being false while its antecedent (''x'' —< ''y'') —< ''x'' is true. If this is true, either its consequent, ''x'', is true, when the the whole formula would be true, or its antecedent ''x'' —< ''y'' is false. But in the last case the antecedent of ''x'' —< ''y'', that is ''x'', must be true. (Peirce, CP 3.384).
Peirce goes on to point out an immediate application of the law:
: From the formula just given, we at once get:
{| align="center"
| {(''x'' —< ''y'') —< ''a''} —< ''x'',
|}
: where the ''a'' is used in such a sense that (''x'' —< ''y'') —< ''a'' means that from (''x'' —< ''y'') every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of ''x'' follows the truth of ''x''. (Peirce, CP 3.384).
=Relation in TeX=
A '''relation''' is a mathematical object of a very general type, the generality of which is best approached in several stages, as will be carried out below. The basic idea, however, is to generalize the concept of a [[binary relation]], such as the binary relations of equality and order that are denoted by the signs "=" and "<" in statements of the form "5 + 7 = 12" and "5 < 12". The concept of a '''relation''' is also the fundamental notion in the [[relational model]] for [[database]]s.
A '''finitary relation''' or a '''polyadic relation''' — specifically a '''k-ary relation''', a '''k-adic relation''', or a '''k-place relation''' when the [[parameter]] ''k'', called the ''[[arity]]'', the ''[[adicity]]'', or the ''[[dimension]]'' of the relation, is known to apply — is conceived according to a formal definition to be given shortly. But it serves understanding to introduce a few preliminary ideas in preparation for the formal definition.
A '''relation''' <math>L</math> is defined by specifying two mathematical objects as its constituent parts:
:* The first part is called the ''frame'' of <math>L</math>, written <math>frame\,(L)</math> or <math>F(L).</math>
:* The second part is called the ''graph'' of <math>L</math>, written <math>graph\,(L)</math> or <math>G(L).</math>
In the special case of a '''finitary relation''', for concreteness a '''k-place relation''', the concepts of its frame and its graph are defined as follows:
:* The '''frame''' of <math>L</math> is specified by giving a [[sequence]] of <math>k</math> [[set]]s, <math>X_1, \ldots , X_k,</math> called the ''domains'' of the relation <math>L,</math> and taking the ''frame'' of <math>L</math> to be their set-theoretic product or [[cartesian product]] <math>F(L) = X_1 \times \ldots \times X_k.</math>
:* The '''graph''' of <math>L</math> is given by specifying a [[subset]] of this cartesian product, and taking the ''graph'' of <math>L</math> to be this subset, <math>G(L) \subseteq F(L) = X_1 \times \ldots \times X_k.</math>
Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''), ''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''), ''G''(''L'')) or the graph ''G''(''L''). There is usually no confusion about this so long as the frame of the relation can be gathered from context.
==Definition==
A '''relation''' ''L'' over the [[set]]s ''X''<sub>1</sub>, …, ''X''<sub>k</sub> is a (''k''+1)-tuple ''L'' = (''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>, ''G''(''L'')) where ''G''(''L'') is a [[subset]] of ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> (the cartesian product of these sets). 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 ''relation over X''. ''G''(''L'') is called the ''graph'' of ''L'' and, as in the case of binary relations, ''L'' is often identified with its graph.
An ''k''-ary '''predicate''' is a [[boolean-valued function]] of ''k'' variables.
==Remarks==
Because a relation as above defines uniquely a k-ary predicate that holds for ''x''<sub>1</sub>, …, ''x''<sub>k</sub> if (''x''<sub>1</sub>, …, ''x''<sub>k</sub>) is in G(''R''), and vice versa, the relation and the predicate are often [[Polymorphism (computer science)|denoted with the same symbol]]. So, for example, the following two statements are considered to be equivalent:
:* <math> (x_1, x_2,\dotsb)\in G(R)</math>
:* <math> R(x_1, x_2,\dotsb)</math>
Relations are classified according to the number of sets in the Cartesian product; in other words the number of terms in the expression:
* Unary relation or [[property (philosophy)|property]]: ''R''(''x'')
* Binary relation: ''R''(''x'', ''y'') or ''x'' ''R'' ''y''
* Ternary relation: ''R''(''x'', ''y'', ''z'')
* Quaternary relation: ''R''(''x'', ''y'', ''z'', ''w'')
Relations with more than 4 terms are usually called ''k''-ary; for example "a 5-ary relation".
==See also==
* [[Binary relation]]
* [[Computable predicate]]
* [[Logic of relatives]]
* [[Theory of relations]]
=Relation in WiX=
A '''relation''' is a mathematical object of a very general type, the generality of which is best approached in several stages, as will be carried out below. The basic idea, however, is to generalize the concept of a [[binary relation]], such as the binary relations of equality and order that are denoted by the signs "=" and "<" in statements of the form "5 + 7 = 12" and "5 < 12". The concept of a '''relation''' is also the fundamental notion in the [[relational model]] for [[database]]s.
A '''finitary relation''' or a '''polyadic relation''' — specifically a '''k-ary relation''', a '''k-adic relation''', or a '''k-place relation''' when the [[parameter]] ''k'', called the ''[[arity]]'', the ''[[adicity]]'', or the ''[[dimension]]'' of the relation, is known to apply — is conceived according to a formal definition to be given shortly. But it serves understanding to introduce a few preliminary ideas in preparation for the formal definition.
A '''relation''' ''L'' is defined by specifying two mathematical objects as its constituent parts:
:* The first part is called the ''frame'' of ''L'', written ''frame''(''L'') or ''F''(''L'').
:* The second part is called the ''graph'' of ''L'', written ''graph''(''L'') or ''G''(''L'').
In the special case of a '''finitary relation''', for concreteness a '''k-place relation''', the concepts of its frame and its graph are defined as follows:
:* The '''frame''' of ''L'' is specified by giving a [[sequence]] of ''k'' [[set]]s, ''X''<sub>1</sub>,…, ''X''<sub>''k''</sub>, called the ''domains'' of the relation ''L'' and taking the ''frame'' of ''L'' to be their set-theoretic product or [[cartesian product]] ''F''(''L'') = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub>.
:* The '''graph''' of ''L'' is given by specifying a [[subset]] of this cartesian product, and taking the ''graph'' of ''L'' to be this subset, ''G''(''L'') ⊆ ''F''(''L'') = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub>.
Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''), ''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''), ''G''(''L'')) or the graph ''G''(''L''). There is usually no confusion about this so long as the frame of the relation can be gathered from context.
==Definition==
A '''relation''' ''L'' over the [[set]]s ''X''<sub>1</sub>, …, ''X''<sub>k</sub> is a (''k''+1)-tuple ''L'' = (''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>, ''G''(''L'')) where ''G''(''L'') is a [[subset]] of ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> (the cartesian product of these sets). 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 ''relation over X''. ''G''(''L'') is called the ''graph'' of ''L'' and, as in the case of binary relations, ''L'' is often identified with its graph.
An ''k''-ary '''predicate''' is a [[boolean-valued function]] of ''k'' variables.
==Remarks==
Because a relation as above defines uniquely a k-ary predicate that holds for ''x''<sub>1</sub>, …, ''x''<sub>k</sub> if (''x''<sub>1</sub>, …, ''x''<sub>k</sub>) is in G(''R''), and vice versa, the relation and the predicate are often [[Polymorphism (computer science)|denoted with the same symbol]]. So, for example, the following two statements are considered to be equivalent:
:* <math> (x_1, x_2,\dotsb)\in G(R)</math>
:* <math> R(x_1, x_2,\dotsb)</math>
Relations are classified according to the number of sets in the Cartesian product; in other words the number of terms in the expression:
* Unary relation or [[property (philosophy)|property]]: ''R''(''x'')
* Binary relation: ''R''(''x'', ''y'') or ''x'' ''R'' ''y''
* Ternary relation: ''R''(''x'', ''y'', ''z'')
* Quaternary relation: ''R''(''x'', ''y'', ''z'', ''w'')
Relations with more than 4 terms are usually called ''k''-ary; for example "a 5-ary relation".
==See also==
* [[Binary relation]]
* [[Computable predicate]]
* [[Logic of relatives]]
* [[Theory of relations]]
=Quote Boxes=
==Format 1==
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
==Format 2==
<blockquote>
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
</blockquote>
=Textbox=
{| class="messagebox standard-talk"
|-
|<center> <font size="+1">Welcome to the discussion</center></font>
|}
<table class="messagebox"><tr><td>
Another night in the box.
</td></tr></table>
<table class="messagebox">
<tr><td>
''f''<math>\;:\;</math>''x''<math>{}\mapsto{}</math>''y'' means ''f''(''x'') = ''y''.
</td></tr>
</table>
<table class="messagebox">
<tr><td>
<math>f\ :\ x\ \to\ y</math> means <math>f\ (x)\ =\ y</math>
</td></tr>
</table>
<table class="messagebox">
<tr><td>
Instead of:
''f''<math>\;:\;</math>''x''<math>{}\mapsto{}</math>''y'' means ''f''(''x'') = ''y''.
Try:
<math>f\ :\ x\ \mapsto\ y</math> means <math>f\ (x)\ =\ y.</math>
</td></tr>
</table>
{| class="messagebox" border="1" cellpadding="6" style="background:#ffdead"
|
The portrait of [[Sojourner Truth]] by [[Norman B. Wood]], entitled ''White Side of a Black Subject'' (1897), is a germane and suitable illustration for the article ''[[Truth theory]]''. The portrait and its subject may be taken to exemplify the self-declared and self-deliberated soul in its journey toward truth. It is contrary to Wikipedia's policy on censorship to remove this image from the article without a compelling reason to do so. Please refer to '''[[WP:NOT#Wikipedia is not censored]]''' for additional information about this policy.
|}
{| align="center" border="1" cellpadding="6" style="background:#ffdead; width:80%"
|
The portrait of [[Sojourner Truth]] by [[Norman B. Wood]], entitled ''White Side of a Black Subject'' (1897), is a germane and suitable illustration for the article ''[[Truth theory]]''. The portrait and its subject may be taken to exemplify the self-declared and self-deliberated soul in its journey toward truth. It is contrary to Wikipedia's policy on censorship to remove this image from the article without a compelling reason to do so. Please refer to '''[[WP:NOT#Wikipedia is not censored]]''' for additional information about this policy.
|}
=Wisdom of Fonts=
<math>\mathcal{A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M}</math>
<math>\mathcal{N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z}</math>
<math>\mathcal{a\,b\,c\,d\,e\,f\,g\,h\,i\,j\,k\,l\,m}</math>
<math>\mathcal{n\,o\,p\,q\,r\,s\,t\,u\,v\,w\,x\,y\,z}</math>
<math>\mathcal{0\,1\,2\,3\,4\,5\,6\,7\,8\,9}</math>
<math>\mathcal{0}</math>
<math>\mathcal{1\ 2\ 3}</math>
<math>\mathcal{4\ 5\ 6}</math>
<math>\mathcal{7\ 8\ 9}</math>
<math>\mathcal{6}</math>
<math>\mathcal{7}</math>
<math>\mathcal{8}</math>
<math>\mathcal{9}</math>
<math>\mathcal{61}</math>
<math>\mathcal{6\,8}</math>
<math>\mathcal{6\,9}</math>
<math>\mathcal{6\,n}</math>
<math>|\!\mathcal{6\,n}\!|</math>
<math>\mathcal{4\,7\,5}</math>
<math>\mathcal{4\ 7\ 7\ 7\ 7\ 7\ 5}</math>
<math>\mathcal{4\,6\,7\,5}</math>
<math>\mathcal{4\,6\,7\,8}</math>
=Joins — Natural Or Else=
<math> -\!< </math>
<math>\triangleright \triangleleft</math>
<math>\triangleright\!\triangleleft</math>
Voila!
<math>\begin{matrix} a & b \\ c & d \end{matrix}</math>
<math>\triangleright\!\triangleleft</math>
<math>\begin{matrix} \triangleright\!\triangleleft \\ R & S \end{matrix}</math>
<math>\begin{matrix} \triangleright\!\triangleleft \\ R & \theta & S \end{matrix}</math>
<math>R \begin{matrix} \triangleright\!\triangleleft \\ i\ \theta\ j \end{matrix} S</math>
<math>\begin{matrix} R\ \triangleright\!\triangleleft\ S \\ i\, \theta\, j \end{matrix}</math>
<math> >< \!</math>
<math> >\!< </math>
<math> |>\!<| </math>
<math> |\!>\!<\!| </math>
<math> |\!>\!<\!| </math>
<math>\begin{matrix}R\ \triangleright\!\triangleleft\ S \\ \ i\ \theta\ j\end{matrix}</math>
<math>\begin{matrix}R\ |\!>\!<\!|\ S \\ i\ \theta\ j\end{matrix}</math>
=Eunucode=
: The entity named nbsp is a '''n'''on-'''b'''reaking '''sp'''ace, so a formula or equation will not have an awkward line break appear in its midst. An alternative is to paste in a UTF-8 unicode character like thinsp, which should appear as whitespace in the edit window, and (since it is not the "space" character) also prevent line breaking: ''a'' = ''b''. Here's a list of sample spacing options: ensp (" "), emsp (" "), emsp13 (" "), emsp14 (" "), numsp (" "), puncsp (" "), thinsp (" "), VeryThinSpace (" "). --[[User:KSmrq|KSmrq]]<sup>[[User talk:KSmrq|T]]</sup> 06:35, 3 February 2006 (UTC)
=Junkyard=
<math>proj_{XY}(L) = L_{XY} = \{(x, y) \in X \times Y : (\exists z \in Z) (x, y, z) \in L \}</math>
<math>proj_{XZ}(L) = L_{XZ} = \{(x, z) \in X \times Z : (\exists y \in Y) (x, y, z) \in L \}</math>
<math>proj_{YZ}(L) = L_{YZ} = \{(y, z) \in Y \times Z : (\exists x \in X) (x, y, z) \in L \}</math>
<math>proj_{XY}(L) = L_{XY} = \{(x, y) \in X \times Y \mid \exists z \in Z \mid (x, y, z) \in L \}</math>
<math>proj_{XZ}(L) = L_{XZ} = \{(x, z) \in X \times Z \mid \exists y \in Y \mid (x, y, z) \in L \}</math>
<math>proj_{YZ}(L) = L_{YZ} = \{(y, z) \in Y \times Z \mid \exists x \in X \mid (x, y, z) \in L \}</math>
==QV Table==
{| style="width:90%"
| '''Elements''' || ||
|-
| [[Attribute]]
| [[Distinctive feature]]
| [[Feature]]
|-
| [[Function (mathematics)|Function]]
| [[Functional (mathematics)|Functional]]
| [[Quality]]
|-
| '''Algebra''' || ||
|-
| [[Category theory]]
| [[Operation (mathematics)|Operation]]
| [[Operator]]
|-
| [[Multigrade operator]]
| [[Parametric operator]]
|
|-
| [[Relation algebra]]
| [[Universal algebra]]
|
|-
| '''Combinatorics, geometry, set theory''' || ||
|-
| [[Relation (mathematics)|Relation]]
| [[Relation composition]]
|
|-
| [[Relation construction]]
| [[Relation reduction]]
| [[Theory of relations]]
|}
'''Logic'''
{{col-begin}}
{{col-break}}
* [[Predicate]]
* [[Property (philosophy)|Property]]
{{col-break}}
* [[Proposition]]
* [[Relative term]]
{{col-break}}
* [[Rheme]]
* [[Role]]
{{col-end}}
'''Computer science'''
{{col-begin}}
{{col-break}}
* [[Database]]
* [[Relational algebra]]
{{col-break}}
* [[Relational database]]
* [[Relational model]]
{{col-end}}
'''Primary sources'''
* [[Logic of Relatives (1870)]]
* [[Logic of Relatives (1883)]]
'''Elements'''
{{col-begin}}
{{col-break}}
* [[Attribute]]
* [[Distinctive feature]]
{{col-break}}
* [[Feature]]
* [[Function (mathematics)|Function]]
{{col-break}}
* [[Functional (mathematics)|Functional]]
* [[Quality]]
{{col-end}}
'''Algebra'''
{{col-begin}}
{{col-break}}
* [[Category theory]]
* [[Operation (mathematics)|Operation]]
* [[Operator]]
{{col-break}}
* [[Multigrade operator]]
* [[Parametric operator]]
{{col-break}}
* [[Relation algebra]]
* [[Universal algebra]]
{{col-end}}
'''Combinatorics, geometry, set theory'''
{{col-begin}}
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
{{col-break}}
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Theory of relations]]
{{col-end}}
'''Logic'''
{{col-begin}}
{{col-break}}
* [[Predicate]]
* [[Property (philosophy)|Property]]
{{col-break}}
* [[Proposition]]
* [[Relative term]]
{{col-break}}
* [[Rheme]]
* [[Role]]
{{col-end}}
'''Computer science'''
{{col-begin}}
{{col-break}}
* [[Database]]
* [[Relational algebra]]
{{col-break}}
* [[Relational database]]
* [[Relational model]]
{{col-end}}
'''Primary sources'''
* [[Logic of Relatives (1870)]]
* [[Logic of Relatives (1883)]]
=Casing the Joint=
<math>
f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}
</math>
<math>
\psi_{\mbox{CIRCLE}}(X) =
\begin{cases} 1 & \mbox{if the figure }X \mbox{ is a circle,}
\\ 0 & \mbox{if the figure is not a circle.}
\end{cases}
</math>
==New Biz==
===Typographical Towers===
'''Version 1'''
'''Example 1'''. Any algebra being trivially a homologue of itself, the algebra of finitary operations on {0, 1} qualifies as a Boolean algebra. To understand the operations of Boolean algebra and their laws in general it therefore suffices to understand them for just this two-element Boolean algebra.
There being ''k''<sup>''k''</sup><sup><sup>''n''</sup></sup> ''n''-ary operations ''f'': ''X''<sup>''n''</sup>→''X'' on a ''k''-element set ''X'', there are therefore 2<sup>2</sup><sup><sup>''n''</sup></sup> ''n''-ary operations on {0,1}. Although we don't need to specify an order for the operations, it is natural to list the smaller arities first. This then makes the signature of a Boolean algebra
: 0-0-1-1-1-1-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-3-3-3-…,
meaning that every Boolean algebra, however small or large, has two constants or "nullary" operations, four unary operations, 16 binary operations, 256 ternary, etc., which we call the '''[[Boolean operation]]s''' of the given Boolean algebra.
'''Version 2'''
'''Example 1'''. Any algebra being trivially a homologue of itself, the algebra of finitary operations on {0, 1} qualifies as a Boolean algebra. To understand the operations of Boolean algebra and their laws in general it therefore suffices to understand them for just this two-element Boolean algebra.
There being ''k''<sup>''k''</sup><sup><sup>''n''</sup></sup> ''n''-ary operations ''f'': ''X''<sup>''n''</sup>→''X'' on a ''k''-element set ''X'', there are therefore 2<sup>2</sup><sup><sup>''n''</sup></sup> ''n''-ary operations on {0,1}. Although we don't need to specify an order for the operations, it is natural to list the smaller arities first. This then makes the signature of a Boolean algebra
: 0-0-1-1-1-1-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-3-3-3-…,
meaning that every Boolean algebra, however small or large, has two constants or "nullary" operations, four unary operations, 16 binary operations, 256 ternary, etc., which we call the '''[[Boolean operation]]s''' of the given Boolean algebra.
===Themes and variations===
{| class="wikitable" align="center" style="text-align:center; width:90%"
|-
! Laws \ Explananda !! Particular Facts !! General Regularities
|-
! Universal Laws
| D-N <p> Deductive-Nomological || D-N <p> Deductive-Nomological
|-
! Statistical Laws
| I-S <p> Inductive-Statistical || D-S <p> Deductive-Statistical
|}
{| class="wikitable"
|+Multiplication table
|-
! × !! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|-
! 4
| 4 || 8 || 12
|-
! 5
| 5 || 10 || 15
|}
{| class="wikitable"
|+Multiplication table
|-
! × !! 1 !! 2 !! 3
|-
! 1
| 1 || 2 || 3
|-
! 2
| 2 || 4 || 6
|-
! 3
| 3 || 6 || 9
|-
! 4
| 4 || 8 || 12
|-
! 5
| 5 || 10 || 15
|}
===Truth and its vicissitudes===
====Vicissitude 1====
'''''Truth''''' ([[antonym]] ''[[false|falsity]]'') refers to the property of a [[proposition]] or its [[symbol]]ic expression having a degree of fidelity with ''[[reality]]''. A statement that is judged to have the property of truth is said to be ''true'', and may be referred to in substantive terms as "a truth". The [[hypostatic object|abstract object]] to which all true statements may be taken to refer is frequently referred to in general terms as "the truth". In [[rhetoric]]al contexts where [[obfuscation]] is a factor, ''[[honesty]]'' and ''[[sincerity]]'' may also be considered as aspects of the "truth".
====Vicissitude 2====
'''''Truth''''' ([[antonym|opposite]] ''[[false|falsity]]'') refers to the property of a [[proposition]] or its [[symbol]]ic expression as having a strong fidelity with ''[[reality]]''. A statement that is judged to have the property of truth is said to be ''true'', and may be referred to in substantive terms as "a truth". The [[hypostatic object|abstract object]] to which all true statements may be taken to refer is also referred to in general terms as "the truth". In [[rhetoric]]al contexts where [[obfuscation]] is a factor, ''[[honesty]]'' and ''[[sincerity]]'' may also be considered as aspects of the "truth".
==Old Biz==
{| cellpadding="2"
|-
| style="width:20px" |
| align="center" | <math>(G \circ H)_{ij}</math>
|
|-
|
| align="center" | <math>=\!</math>
| the <math>ij</math><sup>th</sup> entry in the matrix representation for <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the entry in the <math>i</math><sup>th</sup> row and the <math>j</math><sup>th</sup> column of <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the scalar product of the <math>i</math><sup>th</sup> row of <math>G\!</math> with the <math>j</math><sup>th</sup> column of <math>H\!</math>
|-
|
| align="center" | <math>=\!</math>
| <math>\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}</math>
|}
{| border="1" cellpadding="4"
|-
| style="width:20px" |
| align="center" | <math>(G \circ H)_{ij}</math>
|
|-
|
| align="center" | <math>=\!</math>
| the <math>ij</math><sup>th</sup> entry in the matrix representation for <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the entry in the <math>i</math><sup>th</sup> row and the <math>j</math><sup>th</sup> column of <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the scalar product of the <math>i</math><sup>th</sup> row of <math>G\!</math> with the <math>j</math><sup>th</sup> column of <math>H\!</math>
|-
|
| align="center" | <math>=\!</math>
| <math>\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}</math>
|}
{| border="1" cellspacing="4"
|-
| style="width:20px" |
| align="center" | <math>(G \circ H)_{ij}</math>
|
|-
|
| align="center" | <math>=\!</math>
| the <math>ij</math><sup>th</sup> entry in the matrix representation for <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the entry in the <math>i</math><sup>th</sup> row and the <math>j</math><sup>th</sup> column of <math>G \circ H</math>
|-
|
| align="center" | <math>=\!</math>
| the scalar product of the <math>i</math><sup>th</sup> row of <math>G\!</math> with the <math>j</math><sup>th</sup> column of <math>H\!</math>
|-
|
| align="center" | <math>=\!</math>
| <math>\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}</math>
|}
The formula for computing ''G'' o ''H'' says the following:
(G o H)_ij
= the ij^th entry in the matrix representation for G o H
= the entry in the i^th row and the j^th column of G o H
= the scalar product of the i^th row of G with the j^th column of H
= Sum_k (G_ik H_kj)
{|
|-
| align="center" | <math>(G \circ H)_{ij}</math>
|
|
|-
|
| <math>=\;</math>
| the ij^th entry in the matrix representation for G o H
|-
|
| <math>=\;</math>
| the entry in the i^th row and the j^th column of G o H
|-
|
| <math>=\;</math>
| the scalar product of the i^th row of G with the j^th column of H
|-
|
| <math>=\;</math>
| Sum_k (G_ik H_kj)
|}
: <math>(G \circ H)_{ij}</math>
{|
|-
| style="width:20px" |
| =
| the ij^th entry in the matrix representation for G o H
|-
|
| =
| the entry in the i^th row and the j^th column of G o H
|-
|
| =
| the scalar product of the i^th row of G with the j^th column of H
|-
|
| =
| <math>\sum_{k} (G_{ik} H_{kj})</math>
|}
==Matrix Matters==
'''Table Format'''
{|
|-
| style="width:20px" |
| <math>F\ </math>
| <math>=\ 4:3:4</math>
| <math>+\ 4:4:4</math>
| <math>+\ 4:5:4</math>
|-
|
| <math>G\ </math>
| <math>=\ 4:3</math>
| <math>+\ 4:4</math>
| <math>+\ 4:5</math>
|-
|
| <math>H\ </math>
| <math>=\qquad\!\! 3:4</math>
| <math>+\qquad\!\! 4:4</math>
| <math>+\qquad\!\! 5:4</math>
|}
'''Matrix Format'''
: <math>\begin{matrix}
F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\
G & = & 4:3 & + & 4:4 & + & 4:5 \\
H & = & 3:4 & + & 4:4 & + & 5:4
\end{matrix}</math>
==Minimal Negation Operators==
:: ( ) = 0
:: (''x'') = ~''x'' = ¬''x'' = ''x''′
:: (''x'', ''y'') = ''x'' + ''y'' = ''x''′''y'' + ''xy''′
:: (''x'', ''y'', ''z'') = ''x''′''yz'' + ''xy''′''z'' + ''xyz''′
: <math>\begin{matrix}
(\ ) & = & 0 & = & \mbox{false} \\
(x) & = & \neg x & = & \tilde{x} & = & x' \\
(x, y) & = & x + y & = & \tilde{x} y \lor x \tilde{y} & = & x'y \lor xy'
\end{matrix}</math>
==Nested Tables==
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
{| border=1
| α
| align="center" | cell2
{| border=2 style="background-color:#ABCDEF;"
| NESTED
|-
| TABLE
|}
| valign="bottom" | the original table again
|}
{| border=1
| α
| align="center" | cell2
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_0 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 0\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
| valign="bottom" | the original table again
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_0 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 0\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_1 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 1\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
==Relational Data==
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f9f9f9; font-weight:bold; text-align:center; width:90%"
|+ '''Table 1. Relational Database'''
|- style="background:#efefef"
! style="width:15%" | Domain 1
! style="width:15%" | Domain 2
! style="width:15%" | ...
! style="width:15%" | Domain j
! style="width:15%" | ...
! style="width:15%" | Domain k
|-
| x<sub>11</sub>
| x<sub>12</sub>
| ...
| x<sub>1j</sub>
| ...
| x<sub>1k</sub>
|-
| x<sub>21</sub>
| x<sub>22</sub>
| ...
| x<sub>2j</sub>
| ...
| x<sub>2k</sub>
|-
| ... || ... || ... || ... || ... || ...
|-
| x<sub>i1</sub>
| x<sub>i2</sub>
| ...
| x<sub>ij</sub>
| ...
| x<sub>ik</sub>
|-
| ... || ... || ... || ... || ... || ...
|-
| x<sub>m1</sub>
| x<sub>m2</sub>
| ...
| x<sub>mj</sub>
| ...
| x<sub>mk</sub>
|}
==Dyadic Projections==
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OS</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>OI</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
| align="center" style="width:45%" | ''proj''<sub>SI</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
==Projective reducibility of triadic relations==
{{main|Triadic relation}}
By way of illustrating the different sorts of things that can occur in considering the projective reducibility of relations, it is convenient to reuse the four examples of 3-adic relations that are discussed in the main article on that subject.
===Examples of projectively irreducible relations===
The 3-adic relations '''L'''<sub>0</sub> and '''L'''<sub>1</sub> are shown in the next two Tables:
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_0 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 0\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''0'''
|-
| '''0''' || '''1''' || '''1'''
|-
| '''1''' || '''0''' || '''1'''
|-
| '''1''' || '''1''' || '''0'''
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_1 = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 1\}</math>
! X !! Y !! Z
|-
| '''0''' || '''0''' || '''1'''
|-
| '''0''' || '''1''' || '''0'''
|-
| '''1''' || '''0''' || '''0'''
|-
| '''1''' || '''1''' || '''1'''
|}
A ''2-adic projection'' of a 3-adic relation ''L'' is the 2-adic relation that results from deleting one column of the table for ''L'' and then deleting all but one row of any resulting rows that happen to be identical in content. In other words, the multiplicity of any repeated row is ignored.
In the case of the above two relations, '''L'''<sub>0</sub>, '''L'''<sub>1</sub> ⊆ ''X'' × ''Y'' × ''Z'' <u>≈</u> '''B'''<sup>3</sup>, the 2-adic projections are indexed by the columns or domains that remain, as shown in the following Tables.
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''1'''
|-
| '''1''' || '''0'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>0</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|}
|}
{| align="center" style="width:90%"
| align="center" | proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Y
|-
| '''0''' || '''0'''
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! X !! Z
|-
| '''0''' || '''1'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''0'''
|-
| '''1''' || '''1'''
|}
| align="center" | proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! Y !! Z
|-
| '''0''' || '''1'''
|-
| '''1''' || '''0'''
|-
| '''0''' || '''0'''
|-
| '''1''' || '''1'''
|}
|}
It is clear by inspection that the following equations hold:
{| align="center" cellpadding="4" style="width:90%"
| proj<sub>''XY''</sub>('''L'''<sub>0</sub>) = proj<sub>''XY''</sub>('''L'''<sub>1</sub>)
| proj<sub>''XZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''XZ''</sub>('''L'''<sub>1</sub>)
| proj<sub>''YZ''</sub>('''L'''<sub>0</sub>) = proj<sub>''YZ''</sub>('''L'''<sub>1</sub>)
|}
These equations say that '''L'''<sub>0</sub> and '''L'''<sub>1</sub> cannot be distinguished from each other solely on the basis of their 2-adic projection data. In such a case, either relation is said to be ''irreducible with respect to 2-adic projections''. Since reducibility with respect to 2-adic projections is the only interesting case where it concerns the reduction of 3-adic relations, it is customary to say more simply of such a relation that it is ''projectively irreducible'', the 2-adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in non-trivial multiplets of mutually indiscernible relations.
===Examples of projectively reducible relations===
The 3-adic relations '''L'''<sub>A</sub> and '''L'''<sub>B</sub> are shown in the next two Tables:
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_A = \mbox{Sign Relation of Interpreter A}</math>
|-
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"i"'''
|-
| '''A''' || '''"i"''' || '''"A"'''
|-
| '''A''' || '''"i"''' || '''"i"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"u"'''
|-
| '''B''' || '''"u"''' || '''"B"'''
|-
| '''B''' || '''"u"''' || '''"u"'''
|}
{| align="center" class="wikitable" style="text-align:center; width:60%"
|+ <math>\mathbf{L}_B = \mbox{Sign Relation of Interpreter B}</math>
|-
! style="width:20%" | Object
! style="width:20%" | Sign
! style="width:20%" | Interpretant
|-
| '''A''' || '''"A"''' || '''"A"'''
|-
| '''A''' || '''"A"''' || '''"u"'''
|-
| '''A''' || '''"u"''' || '''"A"'''
|-
| '''A''' || '''"u"''' || '''"u"'''
|-
| '''B''' || '''"B"''' || '''"B"'''
|-
| '''B''' || '''"B"''' || '''"i"'''
|-
| '''B''' || '''"i"''' || '''"B"'''
|-
| '''B''' || '''"i"''' || '''"i"'''
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"i"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"u"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>A</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"i"'''
|-
| '''"i"''' || '''"A"'''
|-
| '''"i"''' || '''"i"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"u"'''
|-
| '''"u"''' || '''"B"'''
|-
| '''"u"''' || '''"u"'''
|}
|}
{| align="center" style="width:90%"
| align="center" style="width:30%" | proj<sub>''XY''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Sign
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''XZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Object
! style="width:50%" | Interpretant
|-
| '''A''' || '''"A"'''
|-
| '''A''' || '''"u"'''
|-
| '''B''' || '''"B"'''
|-
| '''B''' || '''"i"'''
|}
| align="center" style="width:30%" | proj<sub>''YZ''</sub>('''L'''<sub>B</sub>)
{| class="wikitable" style="text-align:center; width:90%"
! style="width:50%" | Sign
! style="width:50%" | Interpretant
|-
| '''"A"''' || '''"A"'''
|-
| '''"A"''' || '''"u"'''
|-
| '''"u"''' || '''"A"'''
|-
| '''"u"''' || '''"u"'''
|-
| '''"B"''' || '''"B"'''
|-
| '''"B"''' || '''"i"'''
|-
| '''"i"''' || '''"B"'''
|-
| '''"i"''' || '''"i"'''
|}
|}
==Epigraphs==
===Epigraph 1===
{|
| style="width:44%" |
| ''All rising to Great Place is by a Winding Staire''
|-
| style="width:44%" |
| Francis Bacon, ''Essays, Civil and Moral'' (1625)
|}
===Epigraph 2===
{|
| style="width:44%" |
| ''Hit's a-comin', boys. Tell yore folks hit's a-comin'.''
|-
| style="width:44%" |
| Thomas Wolfe, ''O Lost, A Story of the Buried Life''
|}
===Epigraph 3===
{|
||
| Ye knowe eek, that in forme of speche is chaunge
|-
||
| With-inne a thousand yeer, and wordes tho
|-
||
| That hadden prys, now wonder nyce and straunge
|-
||
| Us thinketh hem; and yet they spake hem so,
|-
||
| And spedde as wel in love as men now do;
|-
||
| Eek for to winne love in sondry ages,
|-
||
| In sondry londes, sondry been usages.
|-
| style="width:40%" |
| [[Geoffrey Chaucer]], [http://en.wikisource.org/wiki/Troilus_and_Criseyde ''Troilus and Criseyde'' (1385)]
|}
Ye knowe eek, that in forme of speche is chaunge
With-inne a thousand yeer, and wordes tho
That hadden prys, now wonder nyce and straunge
Us thinketh hem; and yet they spake hem so,
And spedde as wel in love as men now do;
Eek for to winne love in sondry ages,
In sondry londes, sondry been usages.
Geoffrey Chaucer, "Troilus and Criseyde", 2.4.22-28 (1385)
http://en.wikisource.org/wiki/Troilus_and_Criseyde:Book_II
===Epigraph 4===
{|
| style="width:44%" |
| Men loven of propre kinde newfangelnesse,
|-
| style="width:44%" |
| As briddes doon that men in cages fede.
|-
| style="width:44%" |
| — Geoffrey Chaucer, "The Squire's Tale"
|}
{|
| style="width:44%" |
| Whan it cam him to purpos for to reste,
|-
|
| I trowe he hadde thilke text in minde,
|-
|
| That 'alle thing, repeiring to his kinde,
|-
|
| Gladeth him-self'; thus seyn men, as I gesse;
|-
|
| Men loven of propre kinde newfangelnesse,
|-
|
| As briddes doon that men in cages fede.
|-
| style="width:44%" |
| — Geoffrey Chaucer, "The Squire's Tale"
|}
==Laws of Form==
===Formal Axioms===
'''Format 1'''
Here is one way of reading the axioms under the entitative interpretation:
:* I<sub>1</sub>. true or true = true.
:* I<sub>2</sub>. not true = false.
:* J<sub>1</sub>. ''a'' or not ''a'' = true.
:* J<sub>2</sub>. [''a'' or ''b''] and [''a'' or ''c''] = ''a'' or [''b'' and ''c''].
Here is one way of reading the axioms under the existential interpretation:
:* I<sub>1</sub>. false and false = false.
:* I<sub>2</sub>. not false = true.
:* J<sub>1</sub>. ''a'' and not ''a'' = false.
:* J<sub>2</sub>. [''a'' and ''b''] or [''a'' and ''c''] = ''a'' and [''b'' or ''c''].
'''Format 2'''
Here is one way of reading the axioms under the entitative interpretation:
{| align="center" cellpadding="4" style="width:80%"
| style="width:10%" | I<sub>1</sub>
| style="width:30%" | true or true
| style="width:10%" | =
| style="width:30%" | true
|-
| I<sub>2</sub>
| not true
| =
| false
|-
| J<sub>1</sub>
| ''a'' or not ''a''
| =
| true
|-
| J<sub>2</sub>.
| [''a'' or ''b''] and [''a'' or ''c'']
| =
| ''a'' or [''b'' and ''c'']
|}
Here is one way of reading the axioms under the existential interpretation:
{| align="center" cellpadding="4" style="width:80%"
| style="width:10%" | I<sub>1</sub>
| style="width:30%" | false and false
| style="width:10%" | =
| style="width:30%" | false
|-
| I<sub>2</sub>
| not false
| =
| true
|-
| J<sub>1</sub>
| ''a'' and not ''a''
| =
| false
|-
| J<sub>2</sub>
| [''a'' and ''b''] or [''a'' and ''c'']
| =
| ''a'' and [''b'' or ''c'']
|}
'''Format 3'''
{{col-begin}}
{{col-2}}
* A
* B
* C
* D
{{col-2}}
* 1
* 2
* 3
* 4
{{col-end}}
===Peirce's Law===
'''Format 1'''
Here is Peirce's own statement of the law:
<blockquote>
A ''fifth icon'' is required for the principle of [[excluded middle]] and other propositions connected with it. One of the simplest formulae of this kind is:
</blockquote>
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
<blockquote>
This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent ''x'' being false while its antecedent (''x'' —< ''y'') —< ''x'' is true. If this is true, either its consequent, ''x'', is true, when the the whole formula would be true, or its antecedent ''x'' —< ''y'' is false. But in the last case the antecedent of ''x'' —< ''y'', that is ''x'', must be true. (Peirce, CP 3.384).
</blockquote>
Peirce goes on to point out an immediate application of the law:
<blockquote>
From the formula just given, we at once get:
</blockquote>
{| align="center"
| {(''x'' —< ''y'') —< ''a''} —< ''x'',
|}
<blockquote>
where the ''a'' is used in such a sense that (''x'' —< ''y'') —< ''a'' means that from (''x'' —< ''y'') every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of ''x'' follows the truth of ''x''. (Peirce, CP 3.384).
</blockquote>
'''Format 2'''
Here is Peirce's own statement of the law:
: A ''fifth icon'' is required for the principle of [[excluded middle]] and other propositions connected with it. One of the simplest formulae of this kind is:
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
: This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent ''x'' being false while its antecedent (''x'' —< ''y'') —< ''x'' is true. If this is true, either its consequent, ''x'', is true, when the the whole formula would be true, or its antecedent ''x'' —< ''y'' is false. But in the last case the antecedent of ''x'' —< ''y'', that is ''x'', must be true. (Peirce, CP 3.384).
Peirce goes on to point out an immediate application of the law:
: From the formula just given, we at once get:
{| align="center"
| {(''x'' —< ''y'') —< ''a''} —< ''x'',
|}
: where the ''a'' is used in such a sense that (''x'' —< ''y'') —< ''a'' means that from (''x'' —< ''y'') every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of ''x'' follows the truth of ''x''. (Peirce, CP 3.384).
=Relation in TeX=
A '''relation''' is a mathematical object of a very general type, the generality of which is best approached in several stages, as will be carried out below. The basic idea, however, is to generalize the concept of a [[binary relation]], such as the binary relations of equality and order that are denoted by the signs "=" and "<" in statements of the form "5 + 7 = 12" and "5 < 12". The concept of a '''relation''' is also the fundamental notion in the [[relational model]] for [[database]]s.
A '''finitary relation''' or a '''polyadic relation''' — specifically a '''k-ary relation''', a '''k-adic relation''', or a '''k-place relation''' when the [[parameter]] ''k'', called the ''[[arity]]'', the ''[[adicity]]'', or the ''[[dimension]]'' of the relation, is known to apply — is conceived according to a formal definition to be given shortly. But it serves understanding to introduce a few preliminary ideas in preparation for the formal definition.
A '''relation''' <math>L</math> is defined by specifying two mathematical objects as its constituent parts:
:* The first part is called the ''frame'' of <math>L</math>, written <math>frame\,(L)</math> or <math>F(L).</math>
:* The second part is called the ''graph'' of <math>L</math>, written <math>graph\,(L)</math> or <math>G(L).</math>
In the special case of a '''finitary relation''', for concreteness a '''k-place relation''', the concepts of its frame and its graph are defined as follows:
:* The '''frame''' of <math>L</math> is specified by giving a [[sequence]] of <math>k</math> [[set]]s, <math>X_1, \ldots , X_k,</math> called the ''domains'' of the relation <math>L,</math> and taking the ''frame'' of <math>L</math> to be their set-theoretic product or [[cartesian product]] <math>F(L) = X_1 \times \ldots \times X_k.</math>
:* The '''graph''' of <math>L</math> is given by specifying a [[subset]] of this cartesian product, and taking the ''graph'' of <math>L</math> to be this subset, <math>G(L) \subseteq F(L) = X_1 \times \ldots \times X_k.</math>
Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''), ''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''), ''G''(''L'')) or the graph ''G''(''L''). There is usually no confusion about this so long as the frame of the relation can be gathered from context.
==Definition==
A '''relation''' ''L'' over the [[set]]s ''X''<sub>1</sub>, …, ''X''<sub>k</sub> is a (''k''+1)-tuple ''L'' = (''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>, ''G''(''L'')) where ''G''(''L'') is a [[subset]] of ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> (the cartesian product of these sets). 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 ''relation over X''. ''G''(''L'') is called the ''graph'' of ''L'' and, as in the case of binary relations, ''L'' is often identified with its graph.
An ''k''-ary '''predicate''' is a [[boolean-valued function]] of ''k'' variables.
==Remarks==
Because a relation as above defines uniquely a k-ary predicate that holds for ''x''<sub>1</sub>, …, ''x''<sub>k</sub> if (''x''<sub>1</sub>, …, ''x''<sub>k</sub>) is in G(''R''), and vice versa, the relation and the predicate are often [[Polymorphism (computer science)|denoted with the same symbol]]. So, for example, the following two statements are considered to be equivalent:
:* <math> (x_1, x_2,\dotsb)\in G(R)</math>
:* <math> R(x_1, x_2,\dotsb)</math>
Relations are classified according to the number of sets in the Cartesian product; in other words the number of terms in the expression:
* Unary relation or [[property (philosophy)|property]]: ''R''(''x'')
* Binary relation: ''R''(''x'', ''y'') or ''x'' ''R'' ''y''
* Ternary relation: ''R''(''x'', ''y'', ''z'')
* Quaternary relation: ''R''(''x'', ''y'', ''z'', ''w'')
Relations with more than 4 terms are usually called ''k''-ary; for example "a 5-ary relation".
==See also==
* [[Binary relation]]
* [[Computable predicate]]
* [[Logic of relatives]]
* [[Theory of relations]]
=Relation in WiX=
A '''relation''' is a mathematical object of a very general type, the generality of which is best approached in several stages, as will be carried out below. The basic idea, however, is to generalize the concept of a [[binary relation]], such as the binary relations of equality and order that are denoted by the signs "=" and "<" in statements of the form "5 + 7 = 12" and "5 < 12". The concept of a '''relation''' is also the fundamental notion in the [[relational model]] for [[database]]s.
A '''finitary relation''' or a '''polyadic relation''' — specifically a '''k-ary relation''', a '''k-adic relation''', or a '''k-place relation''' when the [[parameter]] ''k'', called the ''[[arity]]'', the ''[[adicity]]'', or the ''[[dimension]]'' of the relation, is known to apply — is conceived according to a formal definition to be given shortly. But it serves understanding to introduce a few preliminary ideas in preparation for the formal definition.
A '''relation''' ''L'' is defined by specifying two mathematical objects as its constituent parts:
:* The first part is called the ''frame'' of ''L'', written ''frame''(''L'') or ''F''(''L'').
:* The second part is called the ''graph'' of ''L'', written ''graph''(''L'') or ''G''(''L'').
In the special case of a '''finitary relation''', for concreteness a '''k-place relation''', the concepts of its frame and its graph are defined as follows:
:* The '''frame''' of ''L'' is specified by giving a [[sequence]] of ''k'' [[set]]s, ''X''<sub>1</sub>,…, ''X''<sub>''k''</sub>, called the ''domains'' of the relation ''L'' and taking the ''frame'' of ''L'' to be their set-theoretic product or [[cartesian product]] ''F''(''L'') = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub>.
:* The '''graph''' of ''L'' is given by specifying a [[subset]] of this cartesian product, and taking the ''graph'' of ''L'' to be this subset, ''G''(''L'') ⊆ ''F''(''L'') = ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub>.
Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''), ''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''), ''G''(''L'')) or the graph ''G''(''L''). There is usually no confusion about this so long as the frame of the relation can be gathered from context.
==Definition==
A '''relation''' ''L'' over the [[set]]s ''X''<sub>1</sub>, …, ''X''<sub>k</sub> is a (''k''+1)-tuple ''L'' = (''X''<sub>1</sub>, …, ''X''<sub>''k''</sub>, ''G''(''L'')) where ''G''(''L'') is a [[subset]] of ''X''<sub>1</sub> × … × ''X''<sub>''k''</sub> (the cartesian product of these sets). 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 ''relation over X''. ''G''(''L'') is called the ''graph'' of ''L'' and, as in the case of binary relations, ''L'' is often identified with its graph.
An ''k''-ary '''predicate''' is a [[boolean-valued function]] of ''k'' variables.
==Remarks==
Because a relation as above defines uniquely a k-ary predicate that holds for ''x''<sub>1</sub>, …, ''x''<sub>k</sub> if (''x''<sub>1</sub>, …, ''x''<sub>k</sub>) is in G(''R''), and vice versa, the relation and the predicate are often [[Polymorphism (computer science)|denoted with the same symbol]]. So, for example, the following two statements are considered to be equivalent:
:* <math> (x_1, x_2,\dotsb)\in G(R)</math>
:* <math> R(x_1, x_2,\dotsb)</math>
Relations are classified according to the number of sets in the Cartesian product; in other words the number of terms in the expression:
* Unary relation or [[property (philosophy)|property]]: ''R''(''x'')
* Binary relation: ''R''(''x'', ''y'') or ''x'' ''R'' ''y''
* Ternary relation: ''R''(''x'', ''y'', ''z'')
* Quaternary relation: ''R''(''x'', ''y'', ''z'', ''w'')
Relations with more than 4 terms are usually called ''k''-ary; for example "a 5-ary relation".
==See also==
* [[Binary relation]]
* [[Computable predicate]]
* [[Logic of relatives]]
* [[Theory of relations]]
=Quote Boxes=
==Format 1==
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
==Format 2==
<blockquote>
{| align="center"
| {(''x'' —< ''y'') —< ''x''} —< ''x''.
|}
</blockquote>
=Textbox=
{| class="messagebox standard-talk"
|-
|<center> <font size="+1">Welcome to the discussion</center></font>
|}
<table class="messagebox"><tr><td>
Another night in the box.
</td></tr></table>
<table class="messagebox">
<tr><td>
''f''<math>\;:\;</math>''x''<math>{}\mapsto{}</math>''y'' means ''f''(''x'') = ''y''.
</td></tr>
</table>
<table class="messagebox">
<tr><td>
<math>f\ :\ x\ \to\ y</math> means <math>f\ (x)\ =\ y</math>
</td></tr>
</table>
<table class="messagebox">
<tr><td>
Instead of:
''f''<math>\;:\;</math>''x''<math>{}\mapsto{}</math>''y'' means ''f''(''x'') = ''y''.
Try:
<math>f\ :\ x\ \mapsto\ y</math> means <math>f\ (x)\ =\ y.</math>
</td></tr>
</table>
{| class="messagebox" border="1" cellpadding="6" style="background:#ffdead"
|
The portrait of [[Sojourner Truth]] by [[Norman B. Wood]], entitled ''White Side of a Black Subject'' (1897), is a germane and suitable illustration for the article ''[[Truth theory]]''. The portrait and its subject may be taken to exemplify the self-declared and self-deliberated soul in its journey toward truth. It is contrary to Wikipedia's policy on censorship to remove this image from the article without a compelling reason to do so. Please refer to '''[[WP:NOT#Wikipedia is not censored]]''' for additional information about this policy.
|}
{| align="center" border="1" cellpadding="6" style="background:#ffdead; width:80%"
|
The portrait of [[Sojourner Truth]] by [[Norman B. Wood]], entitled ''White Side of a Black Subject'' (1897), is a germane and suitable illustration for the article ''[[Truth theory]]''. The portrait and its subject may be taken to exemplify the self-declared and self-deliberated soul in its journey toward truth. It is contrary to Wikipedia's policy on censorship to remove this image from the article without a compelling reason to do so. Please refer to '''[[WP:NOT#Wikipedia is not censored]]''' for additional information about this policy.
|}
=Wisdom of Fonts=
<math>\mathcal{A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M}</math>
<math>\mathcal{N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z}</math>
<math>\mathcal{a\,b\,c\,d\,e\,f\,g\,h\,i\,j\,k\,l\,m}</math>
<math>\mathcal{n\,o\,p\,q\,r\,s\,t\,u\,v\,w\,x\,y\,z}</math>
<math>\mathcal{0\,1\,2\,3\,4\,5\,6\,7\,8\,9}</math>
<math>\mathcal{0}</math>
<math>\mathcal{1\ 2\ 3}</math>
<math>\mathcal{4\ 5\ 6}</math>
<math>\mathcal{7\ 8\ 9}</math>
<math>\mathcal{6}</math>
<math>\mathcal{7}</math>
<math>\mathcal{8}</math>
<math>\mathcal{9}</math>
<math>\mathcal{61}</math>
<math>\mathcal{6\,8}</math>
<math>\mathcal{6\,9}</math>
<math>\mathcal{6\,n}</math>
<math>|\!\mathcal{6\,n}\!|</math>
<math>\mathcal{4\,7\,5}</math>
<math>\mathcal{4\ 7\ 7\ 7\ 7\ 7\ 5}</math>
<math>\mathcal{4\,6\,7\,5}</math>
<math>\mathcal{4\,6\,7\,8}</math>
=Joins — Natural Or Else=
<math> -\!< </math>
<math>\triangleright \triangleleft</math>
<math>\triangleright\!\triangleleft</math>
Voila!
<math>\begin{matrix} a & b \\ c & d \end{matrix}</math>
<math>\triangleright\!\triangleleft</math>
<math>\begin{matrix} \triangleright\!\triangleleft \\ R & S \end{matrix}</math>
<math>\begin{matrix} \triangleright\!\triangleleft \\ R & \theta & S \end{matrix}</math>
<math>R \begin{matrix} \triangleright\!\triangleleft \\ i\ \theta\ j \end{matrix} S</math>
<math>\begin{matrix} R\ \triangleright\!\triangleleft\ S \\ i\, \theta\, j \end{matrix}</math>
<math> >< \!</math>
<math> >\!< </math>
<math> |>\!<| </math>
<math> |\!>\!<\!| </math>
<math> |\!>\!<\!| </math>
<math>\begin{matrix}R\ \triangleright\!\triangleleft\ S \\ \ i\ \theta\ j\end{matrix}</math>
<math>\begin{matrix}R\ |\!>\!<\!|\ S \\ i\ \theta\ j\end{matrix}</math>
=Eunucode=
: The entity named nbsp is a '''n'''on-'''b'''reaking '''sp'''ace, so a formula or equation will not have an awkward line break appear in its midst. An alternative is to paste in a UTF-8 unicode character like thinsp, which should appear as whitespace in the edit window, and (since it is not the "space" character) also prevent line breaking: ''a'' = ''b''. Here's a list of sample spacing options: ensp (" "), emsp (" "), emsp13 (" "), emsp14 (" "), numsp (" "), puncsp (" "), thinsp (" "), VeryThinSpace (" "). --[[User:KSmrq|KSmrq]]<sup>[[User talk:KSmrq|T]]</sup> 06:35, 3 February 2006 (UTC)
=Junkyard=
<math>proj_{XY}(L) = L_{XY} = \{(x, y) \in X \times Y : (\exists z \in Z) (x, y, z) \in L \}</math>
<math>proj_{XZ}(L) = L_{XZ} = \{(x, z) \in X \times Z : (\exists y \in Y) (x, y, z) \in L \}</math>
<math>proj_{YZ}(L) = L_{YZ} = \{(y, z) \in Y \times Z : (\exists x \in X) (x, y, z) \in L \}</math>
<math>proj_{XY}(L) = L_{XY} = \{(x, y) \in X \times Y \mid \exists z \in Z \mid (x, y, z) \in L \}</math>
<math>proj_{XZ}(L) = L_{XZ} = \{(x, z) \in X \times Z \mid \exists y \in Y \mid (x, y, z) \in L \}</math>
<math>proj_{YZ}(L) = L_{YZ} = \{(y, z) \in Y \times Z \mid \exists x \in X \mid (x, y, z) \in L \}</math>
==QV Table==
{| style="width:90%"
| '''Elements''' || ||
|-
| [[Attribute]]
| [[Distinctive feature]]
| [[Feature]]
|-
| [[Function (mathematics)|Function]]
| [[Functional (mathematics)|Functional]]
| [[Quality]]
|-
| '''Algebra''' || ||
|-
| [[Category theory]]
| [[Operation (mathematics)|Operation]]
| [[Operator]]
|-
| [[Multigrade operator]]
| [[Parametric operator]]
|
|-
| [[Relation algebra]]
| [[Universal algebra]]
|
|-
| '''Combinatorics, geometry, set theory''' || ||
|-
| [[Relation (mathematics)|Relation]]
| [[Relation composition]]
|
|-
| [[Relation construction]]
| [[Relation reduction]]
| [[Theory of relations]]
|}
'''Logic'''
{{col-begin}}
{{col-break}}
* [[Predicate]]
* [[Property (philosophy)|Property]]
{{col-break}}
* [[Proposition]]
* [[Relative term]]
{{col-break}}
* [[Rheme]]
* [[Role]]
{{col-end}}
'''Computer science'''
{{col-begin}}
{{col-break}}
* [[Database]]
* [[Relational algebra]]
{{col-break}}
* [[Relational database]]
* [[Relational model]]
{{col-end}}
'''Primary sources'''
* [[Logic of Relatives (1870)]]
* [[Logic of Relatives (1883)]]
'''Elements'''
{{col-begin}}
{{col-break}}
* [[Attribute]]
* [[Distinctive feature]]
{{col-break}}
* [[Feature]]
* [[Function (mathematics)|Function]]
{{col-break}}
* [[Functional (mathematics)|Functional]]
* [[Quality]]
{{col-end}}
'''Algebra'''
{{col-begin}}
{{col-break}}
* [[Category theory]]
* [[Operation (mathematics)|Operation]]
* [[Operator]]
{{col-break}}
* [[Multigrade operator]]
* [[Parametric operator]]
{{col-break}}
* [[Relation algebra]]
* [[Universal algebra]]
{{col-end}}
'''Combinatorics, geometry, set theory'''
{{col-begin}}
{{col-break}}
* [[Relation (mathematics)|Relation]]
* [[Relation composition]]
{{col-break}}
* [[Relation construction]]
* [[Relation reduction]]
{{col-break}}
* [[Theory of relations]]
{{col-end}}
'''Logic'''
{{col-begin}}
{{col-break}}
* [[Predicate]]
* [[Property (philosophy)|Property]]
{{col-break}}
* [[Proposition]]
* [[Relative term]]
{{col-break}}
* [[Rheme]]
* [[Role]]
{{col-end}}
'''Computer science'''
{{col-begin}}
{{col-break}}
* [[Database]]
* [[Relational algebra]]
{{col-break}}
* [[Relational database]]
* [[Relational model]]
{{col-end}}
'''Primary sources'''
* [[Logic of Relatives (1870)]]
* [[Logic of Relatives (1883)]]
=Casing the Joint=
<math>
f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}
</math>
<math>
\psi_{\mbox{CIRCLE}}(X) =
\begin{cases} 1 & \mbox{if the figure }X \mbox{ is a circle,}
\\ 0 & \mbox{if the figure is not a circle.}
\end{cases}
</math>
