Let $X = {0,1,2,3,4}$ Draw the graph associated with the $<$ relation on $X$. Should this graph be directed or undirected?
The answer of this question is given, but I am looking for an explanation. Apparently this is the answer:
Since the graph of $<$ is $N$=not symmetric, it should be directed.
And the graph is the given numbers with arrows pointing to those numbers that are higher than itself which I understand. I've read through the chapter in my book and it does not mention anything about direct or symmetry. Can someone give me some help with this?
The problem asks us to consider the relation $<$ as a graph in the sense of a digraph, which is just a fancy word for a binary relation that you use when you want to think in terms of "nodes" and "arrows" rather than ordered pairs. Although digraph stands for "directed graph" let's ignore that because the problem uses "directed" in a slightly different way.
Like you said, the way to think about $<$ as a digraph is that for every pair $(x,y)$ with $x<y$ we draw an arrow from $x$ to $y$. We call the digraph "symmetric" if whenever there is an arrow from a node $x$ to a node $y$ there is also an arrow from $y$ back to $x$. This is just the same as the corresponding relation being symmetric. Clearly this is not the case for the relation $<$ and its associated digraph, so by the problem's terminology we call the digraph "directed" rather than symmetric.
The point is that in a symmetric digraph the arrows all come in pairs—an arrow from $x$ to $y$ is accompanied by an arrow from $y$ to $x$—so we can forget about the direction and just replace the pair of arrows with an edge connecting $x$ and $y$. So a symmetric digraph is equivalent to an ordinary graph. A non-symmetric digraph is one where some of the arrows do not occur in pairs like this; these do not reduce to ordinary graphs and must be considered as digraphs because the direction of arrows is important, so the problem calls them "directed."