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==Examples from mathematics==

==Examples from mathematics==

For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, '''L'''₀ and '''L'''₁, that can be described in the following manner.

For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner.

The first order of business is to define the space in which the relations '''L'''₀ and '''L'''₁ take up residence.  This space is constructed as a 3-fold [[cartesian power]] in the following way.

The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence.  This space is constructed as a 3-fold [[cartesian power]] in the following way.

The '''[[boolean domain]]''' is the set '''B''' = {0,&nbsp;1}.

The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.</math>  The plus sign <math>^{\backprime\backprime} + ^{\prime\prime},</math> used in the context of the boolean domain <math>\mathbb{B},</math> denotes addition mod 2.  Interpreted for logic, this amounts to the same thing as the boolean operation of ''[[exclusive disjunction|exclusive or]]'' or ''not equal to''.

The plus sign "+", used in the context of the boolean domain '''B''', denotes addition mod 2.  Interpreted for logic, this amounts to the same thing as the boolean operation of ''exclusive-or'' or ''not-equal-to''.

The third cartesian power of '''B''' is '''B'''³ = {(''x''₁, ''x''₂, ''x''₃) : ''x''_''j'' in '''B''' for ''j'' = 1, 2, 3}= '''B''' &times; '''B''' &times; '''B'''.

The third cartesian power of <math>\mathbb{B}</math> is <math>\mathbb{B}^3 = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \} = \mathbb{B} \times \mathbb{B} \times \mathbb{B}.</math>

In what follows, the space '''X''' &times; '''Y''' &times; '''Z''' is isomorphic to '''B''' &times; '''B''' &times; '''B''' = '''B'''³.

In what follows, the space <math>X \times Y \times Z</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.</math>

The relation '''L'''₀ is defined as follows:

The relation <math>L_0\!</math> is defined as follows:

: '''L'''₀ = {(''x'', ''y'', ''z'') in '''B'''³ : ''x'' + ''y'' + ''z'' = 0}.

: <math>L_0 = \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.</math>

The relation '''L'''₀ is the set of four triples enumerated here:

The relation <math>L_0\!</math> is the set of four triples enumerated here:

: '''L'''₀ = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}.

: <math>L_0 = \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math>

The relation '''L'''₁ is defined as follows:

The relation <math>L_1\!</math> is defined as follows:

: '''L'''₁ = {(''x'', ''y'', ''z'') in '''B'''³ : ''x'' + ''y'' + ''z'' = 1}.

: <math>L_1 = \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.</math>

The relation '''L'''₁ is the set of four triples enumerated here:

The relation <math>L_1\!</math> is the set of four triples enumerated here:

: '''L'''₁ = {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)}.

: <math>L_1 = \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math>

The triples that make up the relations '''L'''₀ and '''L'''₁ are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows:

The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows:

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