1. Let R be an acyclic relation whose first role is functional (simple uniqueness constraint). Now populate R with a set of facts that includes the n–1 consecutive relationships a1Ra2,…, an–1Ran. Graphically these n–1 relationships form a linear chain with n nodes a1 through an. Since R is acyclic, these n nodes are all distinct from one another. If n < 3, the chain is trivially, strongly intransitive. Now consider the non-trivial case where n ≥ 3, and assume that R is not strongly intransitive. Then it is possible to add another R fact that relates a1 directly to an, as shown in red.
Because of acyclicity, a2 ≠ an, so element a1 is now related directly via R to two distinct elements; but this is impossible because of the uniqueness ...