User:Jon Awbrey/OLDSANDBOX
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 twoelement Boolean algebra.
There being k^{k}^{n} nary operations f: X^{n}→X on a kelement set X, there are therefore 2^{2}^{n} nary 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
 0011112222222222222222333…,
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 operations 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 twoelement Boolean algebra.
There being k^{k}^{n} nary operations f: X^{n}→X on a kelement set X, there are therefore 2^{2}^{n} nary 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
 0011112222222222222222333…,
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 operations of the given Boolean algebra.
Themes and variations
Laws \ Explananda  Particular Facts  General Regularities 

Universal Laws  DN DeductiveNomological 
DN DeductiveNomological 
Statistical Laws  IS InductiveStatistical 
DS DeductiveStatistical 
×  1  2  3 

1  1  2  3 
2  2  4  6 
3  3  6  9 
4  4  8  12 
5  5  10  15 
×  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 falsity) refers to the property of a proposition or its symbolic 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 abstract object to which all true statements may be taken to refer is frequently referred to in general terms as "the truth". In rhetorical contexts where obfuscation is a factor, honesty and sincerity may also be considered as aspects of the "truth".
Vicissitude 2
Truth (opposite falsity) refers to the property of a proposition or its symbolic 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 abstract object to which all true statements may be taken to refer is also referred to in general terms as "the truth". In rhetorical contexts where obfuscation is a factor, honesty and sincerity may also be considered as aspects of the "truth".
Old Biz
\((G \circ H)_{ij}\)  
\(=\!\)  the \(ij\)^{th} entry in the matrix representation for \(G \circ H\)  
\(=\!\)  the entry in the \(i\)^{th} row and the \(j\)^{th} column of \(G \circ H\)  
\(=\!\)  the scalar product of the \(i\)^{th} row of \(G\!\) with the \(j\)^{th} column of \(H\!\)  
\(=\!\)  \(\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}\) 
\((G \circ H)_{ij}\)  
\(=\!\)  the \(ij\)^{th} entry in the matrix representation for \(G \circ H\)  
\(=\!\)  the entry in the \(i\)^{th} row and the \(j\)^{th} column of \(G \circ H\)  
\(=\!\)  the scalar product of the \(i\)^{th} row of \(G\!\) with the \(j\)^{th} column of \(H\!\)  
\(=\!\)  \(\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}\) 
\((G \circ H)_{ij}\)  
\(=\!\)  the \(ij\)^{th} entry in the matrix representation for \(G \circ H\)  
\(=\!\)  the entry in the \(i\)^{th} row and the \(j\)^{th} column of \(G \circ H\)  
\(=\!\)  the scalar product of the \(i\)^{th} row of \(G\!\) with the \(j\)^{th} column of \(H\!\)  
\(=\!\)  \(\begin{matrix} \sum_{k} (G_{ik} H_{kj}) \end{matrix}\) 
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)
\((G \circ H)_{ij}\)  
\(=\;\)  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) 
\[(G \circ H)_{ij}\]
=  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  
=  \(\sum_{k} (G_{ik} H_{kj})\) 
Matrix Matters
Table Format
\(F\ \)  \(=\ 4:3:4\)  \(+\ 4:4:4\)  \(+\ 4:5:4\)  
\(G\ \)  \(=\ 4:3\)  \(+\ 4:4\)  \(+\ 4:5\)  
\(H\ \)  \(=\qquad\!\! 3:4\)  \(+\qquad\!\! 4:4\)  \(+\qquad\!\! 5:4\) 
Matrix Format
\[\begin{matrix} F & = & 4:3:4 & + & 4:4:4 & + & 4:5:4 \\ G & = & 4:3 & + & 4:4 & + & 4:5 \\ H & = & 3:4 & + & 4:4 & + & 5:4 \end{matrix}\]
Minimal Negation Operators
 ( ) = 0
 (x) = ~x = ¬x = x′
 (x, y) = x + y = x′y + xy′
 (x, y, z) = x′yz + xy′z + xyz′
\[\begin{matrix}
(\ ) & = & 0 & = & \mbox{false} \\
(x) & = & \neg x & = & \tilde{x} & = & x' \\
(x, y) & = & x + y & = & \tilde{x} y \lor x \tilde{y} & = & x'y \lor xy'
\end{matrix}\]
Nested Tables
proj_{XY}(L_{0})

proj_{XZ}(L_{0})

proj_{YZ}(L_{0})

proj_{XY}(L_{1})

proj_{XZ}(L_{1})

proj_{YZ}(L_{1})

α  cell2

the original table again 
α  cell2

the original table again 
X  Y  Z 

0  0  0 
0  1  1 
1  0  1 
1  1  0 
X  Y  Z 

0  0  1 
0  1  0 
1  0  0 
1  1  1 
Relational Data
Domain 1  Domain 2  ...  Domain j  ...  Domain k 

x_{11}  x_{12}  ...  x_{1j}  ...  x_{1k} 
x_{21}  x_{22}  ...  x_{2j}  ...  x_{2k} 
...  ...  ...  ...  ...  ... 
x_{i1}  x_{i2}  ...  x_{ij}  ...  x_{ik} 
...  ...  ...  ...  ...  ... 
x_{m1}  x_{m2}  ...  x_{mj}  ...  x_{mk} 
Dyadic Projections
proj_{OS}(L_{A})

proj_{OS}(L_{B})

proj_{OI}(L_{A})

proj_{OI}(L_{B})

proj_{SI}(L_{A})

proj_{SI}(L_{B})

proj_{XY}(L_{A})

proj_{XZ}(L_{A})

proj_{YZ}(L_{A})

proj_{XY}(L_{B})

proj_{XZ}(L_{B})

proj_{YZ}(L_{B})

Projective reducibility of triadic relations
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 3adic relations that are discussed in the main article on that subject.
Examples of projectively irreducible relations
The 3adic relations L_{0} and L_{1} are shown in the next two Tables:
X  Y  Z 

0  0  0 
0  1  1 
1  0  1 
1  1  0 
X  Y  Z 

0  0  1 
0  1  0 
1  0  0 
1  1  1 
A 2adic projection of a 3adic relation L is the 2adic 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_{0}, L_{1} ⊆ X × Y × Z ≈ B^{3}, the 2adic projections are indexed by the columns or domains that remain, as shown in the following Tables.
proj_{XY}(L_{0})

proj_{XZ}(L_{0})

proj_{YZ}(L_{0})

proj_{XY}(L_{1})

proj_{XZ}(L_{1})

proj_{YZ}(L_{1})

It is clear by inspection that the following equations hold:
proj_{XY}(L_{0}) = proj_{XY}(L_{1})  proj_{XZ}(L_{0}) = proj_{XZ}(L_{1})  proj_{YZ}(L_{0}) = proj_{YZ}(L_{1}) 
These equations say that L_{0} and L_{1} cannot be distinguished from each other solely on the basis of their 2adic projection data. In such a case, either relation is said to be irreducible with respect to 2adic projections. Since reducibility with respect to 2adic projections is the only interesting case where it concerns the reduction of 3adic relations, it is customary to say more simply of such a relation that it is projectively irreducible, the 2adic basis being understood. It is immediate from the definition that projectively irreducible relations always arise in nontrivial multiplets of mutually indiscernible relations.
Examples of projectively reducible relations
The 3adic relations L_{A} and L_{B} are shown in the next two Tables:
Object  Sign  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" 
Object  Sign  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" 
proj_{XY}(L_{A})

proj_{XZ}(L_{A})

proj_{YZ}(L_{A})

proj_{XY}(L_{B})

proj_{XZ}(L_{B})

proj_{YZ}(L_{B})

Laws of Form
Formal Axioms
Format 1
Here is one way of reading the axioms under the entitative interpretation:
 I_{1}. true or true = true.
 I_{2}. not true = false.
 J_{1}. a or not a = true.
 J_{2}. [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_{1}. false and false = false.
 I_{2}. not false = true.
 J_{1}. a and not a = false.
 J_{2}. [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:
I_{1}  true or true  =  true 
I_{2}  not true  =  false 
J_{1}  a or not a  =  true 
J_{2}.  [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_{1}  false and false  =  false 
I_{2}  not false  =  true 
J_{1}  a and not a  =  false 
J_{2}  [a and b] or [a and c]  =  a and [b or c] 
Format 3
 A
 B
 C
 D
 1
 2
 3
 4
Peirce's Law
Format 1
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:
{(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:
{(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).
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:
{(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:
{(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 databases.
A finitary relation or a polyadic relation — specifically a kary relation, a kadic relation, or a kplace 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 kplace relation, the concepts of its frame and its graph are defined as follows:
 The frame of \(L\) is specified by giving a sequence of \(k\) sets, \(X_1, \ldots , X_k,\) called the domains of the relation \(L,\) and taking the frame of \(L\) to be their settheoretic product or cartesian product \(F(L) = X_1 \times \ldots \times X_k.\)
 The graph of \(L\) is given by specifying a subset of this cartesian product, and taking the graph of \(L\) to be this subset, \(G(L) \subseteq F(L) = X_1 \times \ldots \times X_k.\)
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 sets X_{1}, …, X_{k} is a (k+1)tuple L = (X_{1}, …, X_{k}, G(L)) where G(L) is a subset of X_{1} × … × X_{k} (the cartesian product of these sets). If all of the X_{j} 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 kary predicate is a booleanvalued function of k variables.
Remarks
Because a relation as above defines uniquely a kary predicate that holds for x_{1}, …, x_{k} if (x_{1}, …, x_{k}) is in G(R), and vice versa, the relation and the predicate are often denoted with the same symbol. So, for example, the following two statements are considered to be equivalent:
 \( (x_1, x_2,\dotsb)\in G(R)\)
 \( R(x_1, x_2,\dotsb)\)
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: 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 kary; for example "a 5ary relation".
See also
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 databases.
A finitary relation or a polyadic relation — specifically a kary relation, a kadic relation, or a kplace 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 kplace 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 sets, X_{1},…, X_{k}, called the domains of the relation L and taking the frame of L to be their settheoretic product or cartesian product F(L) = X_{1} × … × X_{k}.
 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_{1} × … × X_{k}.
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 sets X_{1}, …, X_{k} is a (k+1)tuple L = (X_{1}, …, X_{k}, G(L)) where G(L) is a subset of X_{1} × … × X_{k} (the cartesian product of these sets). If all of the X_{j} 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 kary predicate is a booleanvalued function of k variables.
Remarks
Because a relation as above defines uniquely a kary predicate that holds for x_{1}, …, x_{k} if (x_{1}, …, x_{k}) is in G(R), and vice versa, the relation and the predicate are often denoted with the same symbol. So, for example, the following two statements are considered to be equivalent:
 \( (x_1, x_2,\dotsb)\in G(R)\)
 \( R(x_1, x_2,\dotsb)\)
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: 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 kary; for example "a 5ary relation".
See also
Quote Boxes
Format 1
{(x —< y) —< x} —< x. 
Format 2
{(x —< y) —< x} —< x.
Textbox
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 selfdeclared and selfdeliberated 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
\(\mathcal{A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M}\)
\(\mathcal{N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z}\)
\(\mathcal{a\,b\,c\,d\,e\,f\,g\,h\,i\,j\,k\,l\,m}\)
\(\mathcal{n\,o\,p\,q\,r\,s\,t\,u\,v\,w\,x\,y\,z}\)
\(\mathcal{0\,1\,2\,3\,4\,5\,6\,7\,8\,9}\)
\(\mathcal{0}\)
\(\mathcal{1\ 2\ 3}\)
\(\mathcal{4\ 5\ 6}\)
\(\mathcal{7\ 8\ 9}\)
\(\mathcal{6}\)
\(\mathcal{7}\)
\(\mathcal{8}\)
\(\mathcal{9}\)
\(\mathcal{61}\)
\(\mathcal{6\,8}\)
\(\mathcal{6\,9}\)
\(\mathcal{6\,n}\)
\(\!\mathcal{6\,n}\!\)
\(\mathcal{4\,7\,5}\)
\(\mathcal{4\ 7\ 7\ 7\ 7\ 7\ 5}\)
\(\mathcal{4\,6\,7\,5}\)
\(\mathcal{4\,6\,7\,8}\)
Joins — Natural Or Else
\( \!< \)
\(\triangleright \triangleleft\)
\(\triangleright\!\triangleleft\)
Voila!
\(\begin{matrix} a & b \\ c & d \end{matrix}\)
\(\triangleright\!\triangleleft\)
\(\begin{matrix} \triangleright\!\triangleleft \\ R & S \end{matrix}\)
\(\begin{matrix} \triangleright\!\triangleleft \\ R & \theta & S \end{matrix}\)
\(R \begin{matrix} \triangleright\!\triangleleft \\ i\ \theta\ j \end{matrix} S\)
\(\begin{matrix} R\ \triangleright\!\triangleleft\ S \\ i\, \theta\, j \end{matrix}\)
\( >< \!\)
\( >\!< \)
\( >\!< \)
\( \!>\!<\! \)
\( \!>\!<\! \)
\(\begin{matrix}R\ \triangleright\!\triangleleft\ S \\ \ i\ \theta\ j\end{matrix}\)
\(\begin{matrix}R\ \!>\!<\!\ S \\ i\ \theta\ j\end{matrix}\)
Eunucode
 The entity named nbsp is a nonbreaking space, so a formula or equation will not have an awkward line break appear in its midst. An alternative is to paste in a UTF8 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 (" "). KSmrq^{T} 06:35, 3 February 2006 (UTC)
Junkyard
\(proj_{XY}(L) = L_{XY} = \{(x, y) \in X \times Y : (\exists z \in Z) (x, y, z) \in L \}\)
\(proj_{XZ}(L) = L_{XZ} = \{(x, z) \in X \times Z : (\exists y \in Y) (x, y, z) \in L \}\)
\(proj_{YZ}(L) = L_{YZ} = \{(y, z) \in Y \times Z : (\exists x \in X) (x, y, z) \in L \}\)
\(proj_{XY}(L) = L_{XY} = \{(x, y) \in X \times Y \mid \exists z \in Z \mid (x, y, z) \in L \}\)
\(proj_{XZ}(L) = L_{XZ} = \{(x, z) \in X \times Z \mid \exists y \in Y \mid (x, y, z) \in L \}\)
\(proj_{YZ}(L) = L_{YZ} = \{(y, z) \in Y \times Z \mid \exists x \in X \mid (x, y, z) \in L \}\)
QV Table
Elements  
Attribute  Distinctive feature  Feature 
Function  Functional  Quality 
Algebra  
Category theory  Operation  Operator 
Multigrade operator  Parametric operator  
Relation algebra  Universal algebra  
Combinatorics, geometry, set theory  
Relation  Relation composition  
Relation construction  Relation reduction  Theory of relations 
Logic
Template:ColbreakTemplate:ColbreakTemplate:ColbreakTemplate:ColendComputer scienceCasing the Joint
\( f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases} \)
\( \psi_{\mbox{CIRCLE}}(X) = \begin{cases} 1 & \mbox{if the figure }X \mbox{ is a circle,} \\ 0 & \mbox{if the figure is not a circle.} \end{cases} \)