Question

How can the directed graph representing the symmetric closure of a relation on a finite set be constructed from the directed graph for this relation?

Short answer

The directed graph of the relation\(R\)with any arrows in the opposite direction (of already existing arrows) added.

Step-by-step solution

Given

Directed graph of a relation \(R\).

Concept of relation is Sets

The symmetric closure of\(R\)is the union of the relation\(R\)with its inverse relation\({R^{ - 1}}\).

The inverse relation\({R^{ - 1}}\)is the set\(\{ (b,a)\mid (a,b) \in R\} \)

Find the Inverse and the Symmetric Closure

Directed graph of a relation\(R\).

The inverse relation\({R^{ - 1}}\)is then the directed graph of the relation\(R\)where the direction of all arrows has been reversed.

The symmetric closure of \(R\) is then \(R \cup {R^{ - 1}}\), which is thus the directed graph of the relation \(R\) with any arrows in the opposite direction (of already existing arrows) added.