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

Revision as of 14:22, 25 April 2012

372 bytes added ,  14:22, 25 April 2012
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|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
 
|+ <math>\text{Table 33.1}~~\text{Multiplication Operation of the Group}~V_4</math>
 
|- style="height:50px"
 
|- style="height:50px"
−
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
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| width="20%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
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| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{e}</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{f}</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{g}</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
+
| width="20%" style="border-bottom:1px solid black" | <math>\operatorname{h}</math>
 
|- style="height:50px"
 
|- style="height:50px"
 
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
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{| 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%"
 
{| 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>
+
|+ <math>\text{Table 33.2}~~\text{Regular Representation of the Group}~V_4</math>
 
|- style="height:50px"
 
|- style="height:50px"
−
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>*\!</math>
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| style="border-bottom:1px solid black; border-right:1px solid black" | <math>\text{Element}\!</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{1}</math>
+
| colspan="6" style="border-bottom:1px solid black" | <math>\text{Function as Set of Ordered Pairs of Elements}\!</math>
−
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{r}</math>
  −
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{s}</math>
  −
| width="22%" style="border-bottom:1px solid black" | <math>\operatorname{t}</math>
   
|- style="height:50px"
 
|- style="height:50px"
−
| style="border-right:1px solid black" | <math>\operatorname{1}</math>
+
| width="20%" style="border-right:1px solid black" | <math>\operatorname{e}</math>
−
| <math>\operatorname{1}</math>
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| width="4%"  | <math>\{\!</math>
−
| <math>\operatorname{r}</math>
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| width="16%" | <math>(\operatorname{e}, \operatorname{e}),</math>
−
| <math>\operatorname{s}</math>
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| width="20%" | <math>(\operatorname{f}, \operatorname{f}),</math>
−
| <math>\operatorname{t}</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="height:50px"
−
| style="border-right:1px solid black" | <math>\operatorname{r}</math>
+
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
−
| <math>\operatorname{r}</math>
+
| <math>\{\!</math>
−
| <math>\operatorname{1}</math>
+
| <math>(\operatorname{e}, \operatorname{f}),</math>
−
| <math>\operatorname{t}</math>
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| <math>(\operatorname{e}, \operatorname{e}),</math>
−
| <math>\operatorname{s}</math>
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| <math>(\operatorname{e}, \operatorname{h}),</math>
 +
| <math>(\operatorname{e}, \operatorname{g})</math>
 +
| <math>\}\!</math>
 
|- style="height:50px"
 
|- style="height:50px"
−
| style="border-right:1px solid black" | <math>\operatorname{s}</math>
+
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
−
| <math>\operatorname{s}</math>
+
| <math>\{\!</math>
−
| <math>\operatorname{t}</math>
+
| <math>(\operatorname{e}, \operatorname{g}),</math>
−
| <math>\operatorname{1}</math>
+
| <math>(\operatorname{e}, \operatorname{h}),</math>
−
| <math>\operatorname{r}</math>
+
| <math>(\operatorname{e}, \operatorname{e}),</math>
 +
| <math>(\operatorname{e}, \operatorname{f})</math>
 +
| <math>\}\!</math>
 
|- style="height:50px"
 
|- style="height:50px"
−
| style="border-right:1px solid black" | <math>\operatorname{t}</math>
+
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
−
| <math>\operatorname{t}</math>
+
| <math>\{\!</math>
−
| <math>\operatorname{s}</math>
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| <math>(\operatorname{e}, \operatorname{h}),</math>
−
| <math>\operatorname{r}</math>
+
| <math>(\operatorname{e}, \operatorname{g}),</math>
−
| <math>\operatorname{1}</math>
+
| <math>(\operatorname{e}, \operatorname{f}),</math>
 +
| <math>(\operatorname{e}, \operatorname{e})</math>
 +
| <math>\}\!</math>
 
|}
 
|}
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Table 33.2  Regular Representation of the Group V4
 
Table 33.2  Regular Representation of the Group V4
 
Element Function as Set of Ordered Pairs of Elements
 
Element Function as Set of Ordered Pairs of Elements
−
1 { <1, 1>, <r, r>, <s, s>, <t, t> }
+
e { <e, e>, <f, f>, <g, g>, <h, h> }
−
r { <1, r>, <r, 1>, <s, t>, <t, s> }
+
f { <e, f>, <f, e>, <g, h>, <h, g> }
−
s { <1, s>, <r, t>, <s, 1>, <t, r> }
+
g { <e, g>, <f, h>, <g, e>, <h, f> }
−
t { <1, t>, <r, s>, <s, r>, <t, 1> }
+
h { <e, h>, <f, g>, <g, f>, <h, e> }
 
</pre>
 
</pre>
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Table 33.3  Regular Representation of the Group V4
 
Table 33.3  Regular Representation of the Group V4
 
Element Function as Set of Ordered Pairs of Symbols
 
Element Function as Set of Ordered Pairs of Symbols
−
1   { <"1", "1">, <"r", "r">, <"s", "s">, <"t", "t"> }
+
e   { <"e", "e">, <"f", "f">, <"g", "g">, <"h", "h"> }
−
r   { <"1", "r">, <"r", "1">, <"s", "t">, <"t", "s"> }
+
f   { <"e", "f">, <"f", "e">, <"g", "h">, <"h", "g"> }
−
s   { <"1", "s">, <"r", "t">, <"s", "1">, <"t", "r"> }
+
g   { <"e", "g">, <"f", "h">, <"g", "e">, <"h", "f"> }
−
t   { <"1", "t">, <"r", "s">, <"s", "r">, <"t", "1"> }
+
h   { <"e", "h">, <"f", "g">, <"g", "f">, <"h", "e"> }
 
</pre>
 
</pre>
Jon Awbrey
12,080

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