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Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)

Revision as of 02:56, 26 April 2012

404 bytes removed ,  02:56, 26 April 2012
Undo revision 156529 by Jon Awbrey (Talk) Edit Conflict
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==Discussion==
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==Work Area==
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===Alternate Text===
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 +
A '''semigroup''' consists of a nonempty set with an associative LOC on it.  On formal occasions, a semigroup is introduced by means a formula like <math>X = (X, *),\!</math> interpreted to mean that a semigroup <math>X\!</math> is specified by giving two pieces of data, a nonempty set that conventionally, if somewhat ambiguously, goes under the same name <math>{}^{\backprime\backprime} X {}^{\prime\prime},\!</math> plus an associative binary operation denoted by <math>{}^{\backprime\backprime} * {}^{\prime\prime}.\!</math>  In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the name of the underlying set.  In contexts where more than one semigroup is formed on the same set, one may use notations like <math>X_i = (X, *_i)\!</math> to distinguish them.
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 +
===Additive Presentation===
 +
 
 +
====Version 1====
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: The <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> in a semigroup <math>\underline{X} = (X, +, 0),\!</math> for integer <math>n > 0,\!</math> is notated as <math>nx\!</math> and defined as follows.  Proceeding recursively, for <math>n = 1,\!</math> let <math>1x = x,\!</math> and for <math>n > 1,\!</math> let <math>nx = (n-1)x + x.\!</math>
 +
 
 +
: The <math>n^\text{th}\!</math> '''multiple''' of <math>x\!</math> in a monoid <math>\underline{X} = (X, +, 0),\!</math> for integer <math>n \ge 0,\!</math> is defined the same way for <math>n > 0,\!</math> letting <math>0x = 0\!</math> when <math>n = 0.\!</math>
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 +
: The <math>n^\text{th}\!</math> '''multiple''' of <math>x\!</math> in a group <math>\underline{X} = (X, +, 0),\!</math> for any integer <math>n,\!</math> is defined the same way for <math>n \ge 0,\!</math> letting <math>nx = (-n)(-x)\!</math> for <math>n < 0.\!</math>
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 +
====Version 2====
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: In a semigroup written additively, the <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> is notated as <math>nx\!</math> and defined for every positive integer <math>n\!</math> in the following manner.  Proceeding recursively, let <math>1x = x\!</math> and let <math>nx = (n-1)x + x\!</math> for all <math>n > 1.\!</math>
 +
 
 +
: In a monoid written additively, the multiple <math>nx\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>0x = 0\!</math> and proceeding the same way for <math>n > 0.\!</math>
 +
 
 +
: In a group written additively, the multiple <math>nx\!</math> is defined for every integer <math>n\!</math> by letting <math>nx = (-n)(-x)\!</math> for <math>n < 0\!</math> and proceeding the same way for <math>n \ge 0.\!</math>
 +
 
 +
==Table Work==
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
 +
|+ <math>\text{Table 32.1}~~\text{Scheme of a Group Operation Table}</math>
 +
|- style="height:50px"
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| style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
 +
| style="border-bottom:1px solid black" | <math>x_0\!</math>
 +
| style="border-bottom:1px solid black" | <math>\cdots\!</math>
 +
| style="border-bottom:1px solid black" | <math>x_j\!</math>
 +
| style="border-bottom:1px solid black" | <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_0\!</math>
 +
| <math>x_0 * x_0\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>x_0 * x_j\!</math>
 +
| <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_i\!</math>
 +
| <math>x_i * x_0\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>x_i * x_j\!</math>
 +
| <math>\cdots\!</math>
 +
|- style="height:50px"
 +
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
 +
|+ <math>\text{Table 32.2}~~\text{Scheme of the Regular Ante-Representation}</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_0\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_0 * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_0 * x_j),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| <math>\{\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_i\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_i * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_i * x_j),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:80%"
 +
|+ <math>\text{Table 32.3}~~\text{Scheme of the Regular Post-Representation}</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_0\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_0 * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_j * x_0),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| <math>\{\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>x_i\!</math>
 +
| <math>\{\!</math>
 +
| <math>(x_0 ~,~ x_0 * x_i),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>(x_j ~,~ x_j * x_i),\!</math>
 +
| <math>\cdots\!</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| width="12%" style="border-right:1px solid black" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="22%" | <math>\cdots\!</math>
 +
| width="18%" | <math>\cdots\!</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
 +
| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
 +
| width="20%" | <math>(\operatorname{g}, \operatorname{g}),</math>
 +
| width="16%" | <math>(\operatorname{h}, \operatorname{h})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{f}),</math>
 +
| <math>(\operatorname{f}, \operatorname{e}),</math>
 +
| <math>(\operatorname{g}, \operatorname{h}),</math>
 +
| <math>(\operatorname{h}, \operatorname{g})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{g}),</math>
 +
| <math>(\operatorname{f}, \operatorname{h}),</math>
 +
| <math>(\operatorname{g}, \operatorname{e}),</math>
 +
| <math>(\operatorname{h}, \operatorname{f})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{e}, \operatorname{h}),</math>
 +
| <math>(\operatorname{f}, \operatorname{g}),</math>
 +
| <math>(\operatorname{g}, \operatorname{f}),</math>
 +
| <math>(\operatorname{h}, \operatorname{e})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 33.3}~~\text{Regular Representation of the Group}~V_4</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Symbols}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| width="20%" | <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| width="20%" | <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| width="16%" | <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\{\!</math>
 +
| <math>({}^{\backprime\backprime}\text{e}{}^{\prime\prime}, {}^{\backprime\backprime}\text{h}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{f}{}^{\prime\prime}, {}^{\backprime\backprime}\text{g}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{g}{}^{\prime\prime}, {}^{\backprime\backprime}\text{f}{}^{\prime\prime}),</math>
 +
| <math>({}^{\backprime\backprime}\text{h}{}^{\prime\prime}, {}^{\backprime\backprime}\text{e}{}^{\prime\prime})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 34.1}~~\text{Multiplicative Presentation of the Group}~Z_4(\cdot)</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{a}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{b}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{c}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{a}</math>
 +
| <math>\operatorname{b}</math>
 +
| <math>\operatorname{c}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
 +
| <math>\operatorname{a}</math>
 +
| <math>\operatorname{b}</math>
 +
| <math>\operatorname{c}</math>
 +
| <math>\operatorname{1}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
 +
| <math>\operatorname{b}</math>
 +
| <math>\operatorname{c}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{a}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
 +
| <math>\operatorname{c}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{a}</math>
 +
| <math>\operatorname{b}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 34.2}~~\text{Regular Representation of the Group}~Z_4(\cdot)</math>
 +
|- style="height:50px"
 +
| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
 +
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| width="4%"  | <math>\{\!</math>
 +
| width="16%" | <math>(\operatorname{1}, \operatorname{1}),</math>
 +
| width="20%" | <math>(\operatorname{a}, \operatorname{a}),</math>
 +
| width="20%" | <math>(\operatorname{b}, \operatorname{b}),</math>
 +
| width="16%" | <math>(\operatorname{c}, \operatorname{c})</math>
 +
| width="4%"  | <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{a}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{a}),</math>
 +
| <math>(\operatorname{a}, \operatorname{b}),</math>
 +
| <math>(\operatorname{b}, \operatorname{c}),</math>
 +
| <math>(\operatorname{c}, \operatorname{1})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{b}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{b}),</math>
 +
| <math>(\operatorname{a}, \operatorname{c}),</math>
 +
| <math>(\operatorname{b}, \operatorname{1}),</math>
 +
| <math>(\operatorname{c}, \operatorname{a})</math>
 +
| <math>\}\!</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{c}</math>
 +
| <math>\{\!</math>
 +
| <math>(\operatorname{1}, \operatorname{c}),</math>
 +
| <math>(\operatorname{a}, \operatorname{1}),</math>
 +
| <math>(\operatorname{b}, \operatorname{a}),</math>
 +
| <math>(\operatorname{c}, \operatorname{b})</math>
 +
| <math>\}\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Table 35.1}~~\text{Additive Presentation of the Group}~Z_4(+)</math>
 +
|- style="height:50px"
 +
| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>+\!</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{0}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{2}</math>
 +
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{3}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{0}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{2}</math>
 +
| <math>\operatorname{2}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{3}</math>
 +
| <math>\operatorname{3}</math>
 +
| <math>\operatorname{0}</math>
 +
| <math>\operatorname{1}</math>
 +
| <math>\operatorname{2}</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
<pre>
 +
Table 35.2  Regular Representation of the Group Z4(+)
 +
Element Function as Set of Ordered Pairs of Elements
 +
0 { <0, 0>, <1, 1>, <2, 2>, <3, 3> }
 +
1 { <0, 1>, <1, 2>, <2, 3>, <3, 0> }
 +
2 { <0, 2>, <1, 3>, <2, 0>, <3, 1> }
 +
3 { <0, 3>, <1, 0>, <2, 1>, <3, 2> }
 +
</pre>
Jon Awbrey
12,080

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