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<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> Hi, Seafans.</SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>I suppose that no one
understand what A020955<SPAN style="mso-spacerun: yes">
</SPAN>means.<?xml:namespace prefix = o ns =
"urn:schemas-microsoft-com:office:office" /><o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"></SPAN></SPAN> </P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>So, I explain it a
little.<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>First of all,<SPAN
style="mso-spacerun: yes"> see </SPAN>my home page and click “Free
Class”,<SPAN style="mso-spacerun: yes"> recently </SPAN>I<SPAN
style="mso-spacerun: yes"> rewrote </SPAN>it.<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"></SPAN></SPAN> </P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> <A
href="http://boat.zero.ad.jp/~zbi74583/another02.htm">http://boat.zero.ad.jp/~zbi74583/another02.htm</A></SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"></SPAN></SPAN> </P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>The idea “ma”….an
abbreviation of “make”….is represented<SPAN style="mso-spacerun: yes"> by
</SPAN>a mapping from subsets of a set to members of the
set..<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>Because the idea “ma” has
the same<SPAN style="mso-spacerun: yes"> property </SPAN>as a
mapping.<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"></SPAN></SPAN> </P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>Number of subsets of a set
which has N members is 2^N, and the possibility of images of each subsets is N,
so number of the mapping subsets ->members is N^(2^N).<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>But if the set is a model of
Free Class then the mapping must satisfy some<SPAN
style="mso-spacerun: yes"> conditions.
</SPAN>.<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>They are as
follows.<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"></SPAN></SPAN> </P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>If the set is {0,1,2,3,….n}
then { } ma 0, {0} ma1, {0,1} ma 2, {0,1,2} ma 3,….,{0,1,2,3,….n} ma n
.<o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"></SPAN></SPAN> </P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>So, images of n+1 subsets
are already decided.<SPAN
style="mso-spacerun: yes">
</SPAN><o:p></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN>Hence the number of the
mapping or the number of models which has N members is N^(2^N-n-1)
.</SPAN></P><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt">
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</SPAN></P></SPAN><SPAN lang=EN-US
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face="MS UI Gothic" size=2><SPAN lang=EN-US
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style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt">
Yasutoshi</SPAN></P>
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