Geometric meaning of reflexive and symmetric relations

Question

A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of those ordered pairs $(a,b)$ such that $ a \sim b$.

With this notation explain the geometric meaning of the reflexive and symmetric properties.

Since reflexivity implies the presence of all ordered pairs of the type $(a,a)$, may be the geometric meaning is the straight line passing through $(0,0)$ and $(a,a)$ which is nothing but the line $y=x$.

For symmetric presence of $(a,b)$ implies the presence of $(b,a)$. Is its geometric meaning the straight line joining $(a,b)$ and $(b,a)$??

Thanks for the help!!

Answer 1

Your description of reflexivity is correct.

For symmetry it means that the subset $R$ is "symmetric" around the line $y = x$, this means that for any point $(a, b)\in R$ its mirror point $(b, a)\in R$ (it's the point you get by doing reflection in the line $y=x$), i.e. either none of the two points $(a, b)$ and $(b, a)$ is included in $R$, or both of them are included in $R$.

Thus the "graph" of $R$ is the same as its mirror image when doing reflection in $y=x$.

Answer 2

First, you have to define clearly how you are representing a relation between two real numbers graphically.

For example, if $a \sim b$, are you plotting the point $(a, b)$ on the graph? Then it would make sense to plot each point $(a, a)$ for every $a \in S$ (but it will not make up the entire line $y = x$ unless $S = \mathbb R$). If this is your representation, then the relation is symmetric if whenever $(a, b)$ is plotted, so is $(b, a)$. Similarly, for transitivity, whenever $(a, b)$ and $(b, c)$ are plotted, so is $(a, c)$.

Example:
Each color is a partition (and the ones with more than one element are shown along with their bounding squares).

Joining $(a, b)$ and $(b, a)$ by a straight line (segment) does not make much sense, because the two points are not related, the two elements $a$ and $b$ are. Joining elements by lines works when you have a single point representing every element (instead of a pair of elements). Then you can join the elements by lines or rather, by arrows. This will give you a "graph" (in the graph theoretic sense). So, for an equivalence relation on $S$, you will have a complete graph with $S$ as the vertex set.