There are two ways to think about drawing a picture of a relation on a set .
if 
We can draw relations on , because they are subsets of
-- i.e., subsets of the plane. Actually you've been drawing these pictures since way back.
For example, let's consider the relation given by
if
. The picture looks like:
Here I've put in blue, and its complement
in red.
What does look like?
means
, i.e.
. Here's that, in green.
Notice that and
are general rather different things (that's as we saw before).
To get from
, we swap the roles of
and
. visually, this looks like reflection across the line
. Here's another example, with the line
draw on explicitly:
We can use such a visual depiction to compute the domain and range of a relation: the domain is the "horizontal shadow" and the range is the "vertical shadow:
We can also use such a depiction to check some properties of relations. For example: for to be reflexive on
means every
has
. That is, every pair
. All of the pairs of the form
constitute the diagonal line
. Thus we can check reflexivity by asking: does the relation contain the diagonal line?
Here the relation is reflexive --- it contains the line
--- and the relation
is not.
We can also check symmetry. For to be symmetric means
guarantees
. That is, if we reflect
across the line
, we should land in in
again. That is,
being symmetric means
is symmetric about the line
. Neither
nor
above is symmetric.
Transitivity is a bit more fun to check. Transitivity is relevant to pairs and
both in
. Here we'll check whether
, given by
if
, is transitive. We have
and
, so we check to see if
. Here,
is indicated in red --- it's in
. But we need this to hold for every such pair of points in
.
Unfortunately, it does not:
It's a bit hard to see visually just what transitivity means, but you can see that it's got something to do with right triangles.
if
is a finite set
If is a finite set, we can draw what's called a directed graph, orĀ digraph, of the relation
. A directed graph is a collection of vertices, which we draw as points, and directed edges, which we draw as arrows between the points. For example, let's take the set
and the relation
if
.
The digraph corresponding to this relation is draw like this: we know ,
, and
. So we draw three arrows starting from 2. We also know that
and
, so we draw those:
But that's not all. We also know that , for example. So we need to add arrows that start and end at each vertex:
Observe that divides, as a relation on this set, is transitive: whenever we have and
, we automatically have
. We can recognize this in the digraph by checking that, whenever there are two arrows connected head to tail, the third leg of that triangle is present:
Exercise. What do symmetry and reflexivity look like in the digraph?