Question

Determine whether the relations represented by the directed graphs shown in Exercises 23-25 are reflexive, irreflexive, symmetric, antisymmetric, and/or transitive.

Short answer

Exercise 23 is irreflexive.

Exercise 24 is reflexive, antisymmetric and transitive.

Exercise 25 is irreflexive and antisymmetric.

Step-by-step solution

Given data

Exercises 23-25 are given.

Concept of irreflexive, reflexive, symmetric, anti-symmetric and transitive relation

A relation\(R\)on a set\(A\)is irreflexive if\((a,a) \notin R\)for every\(a \in A\).

A relation\(R\)on a set\(A\)is reflexive if\((a,a) \in R\)for every element\(a \in A\).

A relation\(R\)on a set\(A\)is symmetric if\((b,a) \in R\)whenever\((a,b) \in A\).

A relation\(R\)on a set\(A\)is anti-symmetric if\((b,a) \in R\)and\((a,b) \in R\)implies\({\rm{a}} = {\rm{b}}\).

A relation\(R\)on a set\(A\)is transitive if\((a,b) \in R\)and\((b,c) \in R\)implies\((a,c) \in R\).

The relation is reflexive if there is a loop at each vertex; irreflexive if there are no loops at all;symmetric if edges appear only in anti-parallel pairs (edges from one vertex to a second vertex and from thesecond back to the first); anti-symmetric if there is no pair of antiparallel edges; and transitive if all pathsof length 2 (a pair of edges \((x,y)\) and \((y,z))\) are accompanied by the corresponds to the path of length 1 (theedge \((x,z))\).

Step 3:Determine whether Exercises 23is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive

The relation drawn in Exercise 23 is not reflexive but is irreflexive since there are no loops.

It isnot symmetric, since, for instance, the edge \((a,b)\) is present but the edge \((b,\;a)\)is not present. It is not antisymmetric,since both edges \((b,c)\) and \((c,b)\) are present.

It is not transitive, since the path \((b,c),(c,b)\) from \(b\) to \(b\) isnot accompanied by the edge (\(b\),\(b\)).

Step 4:Determine whether Exercises 24 is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive

The relation drawn in Exercise 24 is reflexive and not irreflexive.

Since, there is a loop at each vertex. It is not symmetric, since, for instance, the edge \((b,a)\) is present but the edge\((a,b)\)is not present.

It is anti-symmetric, since there are no pairs of anti-parallel edges.

It is transitive, since the onlynontrivial path of length 2 is \(bac\), and the edge \((b,c)\) is present.

Step 5:Determine whether Exercises 25 is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive

The relation drawn in Exercise 25 is notreflexive but is irreflexive, since, there are no loops.

It is not symmetric, since, for instance, the edge \((b,\;a)\) is present but not the edge \((a,b)\).

It is anti-symmetric, since there are no pairs of anti-parallel edges.

It is nottransitive, since the edges \((a,c)\) and \((c,d)\) are present, but not \((a,d)\).

Thus, exercise 23 is irreflexive. Exercise 24 is reflexive, antisymmetric and transitive. Exercise 25 is irreflexive and antisymmetric.