Final answer:
A real-world example of a relation whose digraph has no directed cycles is a family tree or an academic prerequisite structure, where parentage or course completion flows in one direction without looping back.
Step-by-step explanation:
To describe a relation in the real world whose digraph has no directed cycles, consider a family tree. In a family tree, an individual A can be a parent to B, and B can be a parent to C. However, A cannot be a parent to C directly in this lineage sequence. This relationship forms a digraph with edges pointing from parent to child but contains no directed cycles because you cannot become your own ancestor.
Another illustrative example would be an academic prerequisite structure where a student must complete course A before taking course B, and course B before taking course C. However, completing course A doesn't mean you've completed course C, thus the prerequisite digraph does not have directed cycles. Directed cycles are not present because you can't take a course before taking it's prerequisite.
In a partial order, each element is related to every other element either directly or indirectly through a series of relationships. For example, consider a relation where each person is related to their parent. In this relation, A is related to B because B is their parent, and B is related to C because C is their parent. However, A and C are not related because there is no direct or indirect relationship between them. This is an example of a relation with no directed cycles.