To either draw the graph or explain why the relationship cannot be represented by a graph:
(a) The set $V = \{1,2,...,9\}$ and the relationship $x ∼ y$ when $x−y$ is a non-zero multiple of $3$.
(b) The set $V = \{1,2,...,9\}$ and the relationship $x ∼ y$ when $y$ is a multiple of $x$.
(c) The set $V = \{1,2,...,9\}$ and the relationship $x ∼ y$ when $0 < |x−y| < 3$.
According to me (a) and (c) will be a graph and (b) will not be a graph.
We know a relation is a graph when it is irreflexive and symmetric. (b) will not be a graph since $x$ is a multiple of $x$ and the relation becomes reflexive.
I did not consider (b) to be a graph as I was considering only simple graphs, with no loops or multi-edges.
The following are the images of the graphs:
Is the solution correct?
All can be represented (so long as we permit a directed graph for part b):
a)
b)
c)