Question

To determine when is \((a,b)\) in \({R^*}\).

Short answer

It is possible to fly from a tob.

Step-by-step solution

Given data

\(R = \{ (a,b)\mid \)There is a direct non-stop airline flight from a to \(b\} \)

\(A = {\rm{ Set of all cities }}\).

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 ( a, c) for which there exists an element b such that \((a,b) \in R\) and \((b,c) \in S\).

\({R^*} = R \cup {R^2} \cup \ldots {R^n}\).

Simplify further

Generalizing part (a) and (b), we get

\({R^j}\)\( = \{ (a,b)\mid \)It is possible to fly from a to b with exactly \(j - 1\) stops in some intermediate cities \(\} \).

\(j \in \{ 1,2,3, \ldots n\} \)

Using, \({R^*} = R \cup {R^2} \cup \ldots {R^n}\), we get

\({R^*} = \{ (a,b)\mid \)It is possible to fly from a to \(b\} \)

It is possible to fly from a tob.