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MyWikiBiz, Author Your Legacy — Friday November 01, 2024
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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 ''[[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 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>
The third cartesian power of <math>\mathbb{B}</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.</math>
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>
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 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:
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:
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
<br>
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
|- style="background:paleturquoise"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:60%"
! X !! Y !! Z
|+ <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}</math>
|- style="background:whitesmoke"
! <math>X\!</math> !! <math>Y\!</math> !! <math>Z\!</math>
|-
|-
| '''0''' || '''0''' || '''0'''
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
|-
|-
| '''0''' || '''1''' || '''1'''
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
|-
|-
| '''1''' || '''0''' || '''1'''
| <math>1\!</math> || <math>0\!</math> || <math>1\!</math>
|-
|-
| '''1''' || '''1''' || '''0'''
| <math>1\!</math> || <math>1\!</math> || <math>0\!</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:60%"
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') ∈ '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
|+ <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}</math>
|- style="background:paleturquoise"
|- style="background:whitesmoke"
! X !! Y !! Z
! <math>X\!</math> !! <math>Y\!</math> !! <math>Z\!</math>
|-
|-
| '''0''' || '''0''' || '''1'''
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
|-
|-
| '''0''' || '''1''' || '''0'''
| <math>0\!</math> || <math>1\!</math> || <math>0\!</math>
|-
|-
| '''1''' || '''0''' || '''0'''
| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
|-
|-
| '''1''' || '''1''' || '''1'''
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
|}
|}
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