There are two ways to think about drawing a picture of a relation on a set .
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, or 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?