A relation

on a set

is a subset of the Cartesian product

. The graph of a relation is a directed graph with vertex set

and edges determined by the ordered pairs in

. This Demonstration lets you explore relations on the set

for

through

. Three specific relations ("divides", "congruent mod 3", and "a + 2b is prime") are included. When using the custom option, a "ghost" of a complete graph on

vertices (with loops) appears. Clicking the ghost edges/loops adds edges/loops to the graph and adds the corresponding ordered pairs to your relation. Clicking an edge a second time changes its direction; clicking a third time makes that edge bidirectional. You can view

(the ordered pairs), the adjacency matrix, or the properties of the relation (reflexive, symmetric, antisymmetric, transitive).