Question

To determine \({{\rm{R}}^*}\).

Short answer

The matrix formed is, \(\left( {\begin{array}{*{20}{l}}1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\end{array}} \right)\).

Step-by-step solution

Given data

Given data, \(R = \{ (1,3),(2,4),(3,1),(3,5),(4,3),(5,1),(5,2)\), and \((5,4)\} \)

\(A = \{ 1,2,3,4,5\} \).

Concept used of closure property

The closure of a relation\(R\)with respect to property\(P\)is the relation obtained by adding the minimum number of ordered pairs to\(R\)to obtain property\(P\).

Represent relation \(R\) by matrix

A relation \(R\) can be represented by the matrix \({M_R} = \left( {{m_{ij}}} \right)\)

\({m_{ij}} = \left\{ {\begin{array}{*{20}{l}}{1{\rm{ if }}\left( {{a_i},{b_j}} \right) \in R}\\{0{\rm{ if }}\left( {{a_i},{b_j}} \right) \notin R}\end{array}} \right.\)

The composite \(S^\circ R\) consists of all ordered pairs \(({\rm{a}},{\rm{c}})\) for which there exists an element b such that \((a,b) \in R\) and \((b,c) \in S\).

The way to form these powers is first to form the matrix representing \(R\), namely \({M_R} = \left( {\begin{array}{*{20}{l}}0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&1&0&0\\1&1&0&1&0\end{array}} \right)\) and then take successive Boolean powers of it to get the matrices representing \({{\rm{R}}^2},{{\rm{R}}^3}\), and so on.

Represent relation \({{\rm{R}}^*}\) by matrix

Finally, for part ( .. we take the join of the matrices representing \(R,{R^2}, \ldots ,{R^5}\)).

Since the matrix is a perfectly goodway to express the relation, we will not list the ordered pairs.

The matrix for \({{\rm{R}}^*}\) is the join of the first matrix displayed above and the answers to parts (a) through (d), namely

\({M_{{R^*}}} = {M_R} \vee M_R^{(2)} \vee M_R^{(3)} \vee M_R^{(4)} \vee M_R^{(5)} = \left( {\begin{array}{*{20}{l}}1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\end{array}} \right)\).