## Relations & Their Relatives : 20

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.

$\begin{array}{l|*{8}{c}} E & \rm{Abe} & \rm{Ben} & \rm{Cai} & \rm{Esa} & \rm{Isa} & \rm{Ish} & \rm{Jac} & \rm{Jos} \\[2pt] \hline \\[2pt] \rm{Abe} & \centerdot &&&&&&&\\[12pt] \rm{Ben} && \centerdot &&&&&&\\[12pt] \rm{Cai} & \rm{Cai:Abe} && \centerdot &&&&&\\[12pt] \rm{Esa} &&&& \centerdot &&& \rm{Esa:Jac} &\\[12pt] \rm{Isa} &&&&& \centerdot &&&\\[12pt] \rm{Ish} &&&&& \rm{Ish:Isa} & \centerdot &&\\[12pt] \rm{Jac} &&&&&&& \centerdot &\\[12pt] \rm{Jos} && \rm{Jos:Ben} &&&&&& \centerdot \end{array}$

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