Question

Let \(R\) be a relation on a set \(A\). Explain how to use the directed graph representing \(R\) to obtain the directed graph representing the inverse relation \({R^{ - 1}}\).

Short answer

All the loops and edges in both directions between distinct vertices are retained from the digraph of \(R\)and the directions of the single edges between distinct vertices are reversed.

Step-by-step solution

Given data

\(R\) be a relation on a set \(A\).

Concept of inverse relation

If\(R\)is a relation on the set\(A\)then the inverse relation is defined as follows:\({R^{ - 1}} = \{ (b,a)\mid (a,b) \in R\} \).

Step 3:Obtain the directed graph represents the inverse relation \({R^{ - 1}}\)

Since the inverse relation consists of all pairs \((b,\;a)\)for which \((a,b)\) is in the original relation, take the digraph for \(R\) and reverse the direction on every edge from above definition.

Thus, in the digraph of \({R^{ - 1}}\), all the loops and edges in both directions between distinct vertices are retained from the digraph of \(R\) and the directions of the single edges between distinct vertices are reversed.