Question

Exercises 9–11 determine whether the relation with the given directed graph is a partial order.

Short answer

The relation of the given directed graph is not a partial ordering.

Step-by-step solution

Given data

The directed graph is ,

Concept used of partially ordered set

A relation\(R\)is a poset if and only if,\((x,x)\)is in\({\rm{R}}\)for all x (reflexivity)

\((x,y)\)and \((y,x)\) in R implies \(x = y\) (anti-symmetry),\((x,y)\) and \((y,z)\) in R implies \((x,z)\) is in \({\rm{R}}\) (transitivity).

Find  if the \(R\) is poset

By observing the given directed graph, it is clear that

i. Self loop is available at each and every vertex. So satisfies the condition \((a,a) \in R\forall a \in A \Rightarrow \) so reflexive.

ii. There is a relation from \(a\) to \(b\) but there is no relation exist from \(b\) to \(a\)

The above relation not satisfies at vertex \(d \Rightarrow \) not anti symmetric.

iii. There is an edge from \(a\) to \(b,b\) to \(d\) but not from \(a\) to \(d\) so not transitive.

Therefore, the relation of the given directed graph is not a partial ordering.