Question

Determine the ordered pairs in the relations represented by the directed graph.

Short answer

The list of ordered pairs in this relation is \(\{ (a,a),(a,b),(b,a),(b,b),(c,a),(c,c),(c,d),(d,d)\} \).

Step-by-step solution

Given data

The directed graph is given.

Concept used

A relation\({\rm{R}}\)from\(A = \left\{ {{a_1},{a_2}, \ldots ,{a_m}} \right\}\)to\(B = \left\{ {{b_1},{b_2}, \ldots ,{b_n}} \right\}\)can be represented by the matrix\({M_R} = \left( {{m_{ij}}} \right)\), where\({m_{ij}} = \left\{ {\begin{array}{*{20}{l}}{1{\rm{ if }}\left( {{a_i},{b_j}} \right) \in R}\\{0{\rm{ if }}\left( {{a_i},{b_j}} \right) \notin R}\end{array}} \right.\)

The relation\({\rm{R}}\)on a set\({\rm{A}}\)can be represented by a directed graph which has the elements of\({\rm{A}}\)as its vertices and the ordered pairs\((a,b)\), where\((a,b) \in R\), as edges.

Step 3:Determine the ordered pairs in the relations represented by the directed graph

In accordance to the above definition, the ordered pairs in the relation are found by list of all the pairs \((x,y)\) for which there is an edge from \(x\) to \(y\) in the directed graph as follows:

Hence, the list of ordered pairs in this relation is\(\{ (a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(d,d)\} \).