Re: Peirce List Discussion • HR
We have been considering special properties that a dyadic relation may have, in particular, the following two symmetry properties.
- A dyadic relation
is symmetric if
being in
is in
- A dyadic relation
The first thing to understand about any symmetry of any relation is that it is a property of the whole relation, the whole set of tuples, not a property of individual tuples.
Many properties of dyadic relations can be made visually evident by arranging their ordered pairs in 2-dimensional arrays. Let’s do this for our initial sample of biblical brothers, using the first three letters of their names as row and column labels.
The relation 
![\begin{array}{l|*{8}{c}} B & \rm{Abe} & \rm{Ben} & \rm{Cai} & \rm{Esa} & \rm{Isa} & \rm{Ish} & \rm{Jac} & \rm{Jos} \\[2pt] \hline \\[2pt] \rm{Abe} & \centerdot && \rm{Abe:Cai} &&&&&\\[12pt] \rm{Ben} && \centerdot &&&&&& \rm{Ben:Jos} \\[12pt] \rm{Cai} & \rm{Cai:Abe} && \centerdot &&&&&\\[12pt] \rm{Esa} &&&& \centerdot &&& \rm{Esa:Jac} &\\[12pt] \rm{Isa} &&&&& \centerdot & \rm{Isa:Ish} &&\\[12pt] \rm{Ish} &&&&& \rm{Ish:Isa} & \centerdot &&\\[12pt] \rm{Jac} &&&& \rm{Jac:Esa} &&& \centerdot &\\[12pt] \rm{Jos} && \rm{Jos:Ben} &&&&&& \centerdot \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7C%2A%7B8%7D%7Bc%7D%7D++B+%26+%5Crm%7BAbe%7D+%26+%5Crm%7BBen%7D+%26+%5Crm%7BCai%7D+%26+%5Crm%7BEsa%7D+%26+%5Crm%7BIsa%7D+%26+%5Crm%7BIsh%7D+%26+%5Crm%7BJac%7D+%26+%5Crm%7BJos%7D+%5C%5C%5B2pt%5D++%5Chline+%5C%5C%5B2pt%5D++%5Crm%7BAbe%7D+%26+%5Ccenterdot+%26%26+%5Crm%7BAbe%3ACai%7D+%26%26%26%26%26%5C%5C%5B12pt%5D++%5Crm%7BBen%7D+%26%26+%5Ccenterdot+%26%26%26%26%26%26+%5Crm%7BBen%3AJos%7D+%5C%5C%5B12pt%5D++%5Crm%7BCai%7D+%26+%5Crm%7BCai%3AAbe%7D+%26%26+%5Ccenterdot+%26%26%26%26%26%5C%5C%5B12pt%5D++%5Crm%7BEsa%7D+%26%26%26%26+%5Ccenterdot+%26%26%26+%5Crm%7BEsa%3AJac%7D+%26%5C%5C%5B12pt%5D++%5Crm%7BIsa%7D+%26%26%26%26%26+%5Ccenterdot+%26+%5Crm%7BIsa%3AIsh%7D+%26%26%5C%5C%5B12pt%5D++%5Crm%7BIsh%7D+%26%26%26%26%26+%5Crm%7BIsh%3AIsa%7D+%26+%5Ccenterdot+%26%26%5C%5C%5B12pt%5D++%5Crm%7BJac%7D+%26%26%26%26+%5Crm%7BJac%3AEsa%7D+%26%26%26+%5Ccenterdot+%26%5C%5C%5B12pt%5D++%5Crm%7BJos%7D+%26%26+%5Crm%7BJos%3ABen%7D+%26%26%26%26%26%26+%5Ccenterdot+++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=-1&c=20201002)
The relation 
![\begin{array}{l|*{8}{c}} E & \rm{Abe} & \rm{Ben} & \rm{Cai} & \rm{Esa} & \rm{Isa} & \rm{Ish} & \rm{Jac} & \rm{Jos} \\[2pt] \hline \\[2pt] \rm{Abe} & \centerdot &&&&&&&\\[12pt] \rm{Ben} && \centerdot &&&&&&\\[12pt] \rm{Cai} & \rm{Cai:Abe} && \centerdot &&&&&\\[12pt] \rm{Esa} &&&& \centerdot &&& \rm{Esa:Jac} &\\[12pt] \rm{Isa} &&&&& \centerdot &&&\\[12pt] \rm{Ish} &&&&& \rm{Ish:Isa} & \centerdot &&\\[12pt] \rm{Jac} &&&&&&& \centerdot &\\[12pt] \rm{Jos} && \rm{Jos:Ben} &&&&&& \centerdot \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7C%2A%7B8%7D%7Bc%7D%7D++E+%26+%5Crm%7BAbe%7D+%26+%5Crm%7BBen%7D+%26+%5Crm%7BCai%7D+%26+%5Crm%7BEsa%7D+%26+%5Crm%7BIsa%7D+%26+%5Crm%7BIsh%7D+%26+%5Crm%7BJac%7D+%26+%5Crm%7BJos%7D+%5C%5C%5B2pt%5D++%5Chline+%5C%5C%5B2pt%5D++%5Crm%7BAbe%7D+%26+%5Ccenterdot+%26%26%26%26%26%26%26%5C%5C%5B12pt%5D++%5Crm%7BBen%7D+%26%26+%5Ccenterdot+%26%26%26%26%26%26%5C%5C%5B12pt%5D++%5Crm%7BCai%7D+%26+%5Crm%7BCai%3AAbe%7D+%26%26+%5Ccenterdot+%26%26%26%26%26%5C%5C%5B12pt%5D++%5Crm%7BEsa%7D+%26%26%26%26+%5Ccenterdot+%26%26%26+%5Crm%7BEsa%3AJac%7D+%26%5C%5C%5B12pt%5D++%5Crm%7BIsa%7D+%26%26%26%26%26+%5Ccenterdot+%26%26%26%5C%5C%5B12pt%5D++%5Crm%7BIsh%7D+%26%26%26%26%26+%5Crm%7BIsh%3AIsa%7D+%26+%5Ccenterdot+%26%26%5C%5C%5B12pt%5D++%5Crm%7BJac%7D+%26%26%26%26%26%26%26+%5Ccenterdot+%26%5C%5C%5B12pt%5D++%5Crm%7BJos%7D+%26%26+%5Crm%7BJos%3ABen%7D+%26%26%26%26%26%26+%5Ccenterdot+++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=-1&c=20201002)
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