Question

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

Short answer

The relation then contains all of the above ordered pairs is\(R = \{ (a,a),(b,b),(c,c),(c,d),(d,c),(d,d)\} \).

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

The directed graph contains three loops at points \(a\), \(b\), \(c\) and \(d\).

If there is a loop at \(x\), then the point (\(x\), \(x)\) is in the relation as follows:

\(\begin{array}{c}(a,a) \in R\\(b,b) \in R\\(d,d) \in R\\(d,d) \in R\end{array}\)

The given directed graph contains an arrow from \(a\) to \(b\), which means that the ordered pair (\(a\), \(b\)) is in the relation \((a,\;b) \in R\).

The given directed graph contains an arrow from \(b\) to \(a\), which means that the ordered pair (\(b\), \(a\)) ) is in the relation \((b,a) \in R\).

The given directed graph contains an arrow from \(c\) to \(d\), which means that the ordered pair (\(a\), \(b\)) is in the relation \((c,d) \in R\).

The given directed graph contains an arrow from \(d\) to \(c\), which means that the ordered pair (\(a\), \(b\)) is in the relation \((d,c) \in R\).

Hence, the relation then contains all of the above ordered pairs is\(R = \{ (a,a),(b,b),(c,c),(c,d),(d,c),(d,d)\} \).