Relations & Their Relatives : 20

Re: Peirce List DiscussionHR

We have been considering special properties that a dyadic relation may have, in particular, the following two symmetry properties.

  • A dyadic relation L is symmetric if (x, y) being in L implies that (y, x) is in L.
  • A dyadic relation L is asymmetric if (x, y) being in L implies that (y, x) is not in L.

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 B indicated by “brother of” is a symmetric relation.  The ordered pairs of B are given below.

\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}

The relation E indicated by “elder brother of” is an asymmetric relation.  The ordered pairs of E are given below.

\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}

This entry was posted in C.S. Peirce, Dyadic Relations, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Relation Theory, Semiotics, Sign Relations, Triadic Relations and tagged , , , , , , , , , , . Bookmark the permalink.

2 Responses to Relations & Their Relatives : 20

  1. Pingback: Survey of Relation Theory • 2 | Inquiry Into Inquiry

  2. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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