Changes

Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}.\!</math>

Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}.\!</math>

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| align="center" |

 

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Table 1.  Permutations or Substitutions in Sym {A, B, C}

|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>

o---------o---------o---------o---------o---------o---------o

|- style="background:#f0f0ff"

|         |         |         |         |         |        |

| width="16%" | <math>\operatorname{e}</math>

|   e    |   f    |   g    |   h    |   i    |   j   |

| width="16%" | <math>\operatorname{f}</math>

|         |        |        |        |        |        |

| width="16%" | <math>\operatorname{g}</math>

| width="16%" | <math>\operatorname{h}</math>

| width="16%" | <math>\operatorname{i}</math>

|  A B C | A B C |  A B C  | A B C |  A B C | A B C |

| width="16%" | <math>\operatorname{j}</math>

|-

|

|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |

|  A B C |  C A B |  B C A | A C B |  C B A |  B A C |

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\mathrm{A} & \mathrm{B} & \mathrm{C}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{C} & \mathrm{A} & \mathrm{B}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{B} & \mathrm{C} & \mathrm{A}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

|

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{C} & \mathrm{B} & \mathrm{A}

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\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{B} & \mathrm{A} & \mathrm{C}

\end{matrix}</math>

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Then we rewrote these permutations &mdash; being functions <math>f : X \to X</math> they can also be recognized as being 2-adic relations <math>f \subseteq X \times X</math> &mdash; in ''relative form'', in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance:

Then we rewrote these permutations &mdash; being functions <math>f : X \to X</math> they can also be recognized as being 2-adic relations <math>f \subseteq X \times X</math> &mdash; in ''relative form'', in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance: