Question

Show that if \({{\bf{M}}_R}\) is the matrix representing the relation \(R\), then \({\bf{M}}_R^{(n)}\) is the matrix representing the relation \({R^n}\).

Short answer

The \({\bf{M}}_R^{(n)}\) is the matrix represents the relation \({R^n}\).

Step-by-step solution

Given data

\({{\rm{M}}_{\rm{R}}}\) is the matrix represents the relation \(R\).

Concept used

The matrix represent the composite of two relations can be used to find the matrix for\({M_{Rn}}\)as follows:\({M_{Rn}} = {\bf{M}}_R^{(n)}\)

Step 3:Prove that the \({\bf{M}}_R^{(n)}\)is the matrix represents the relation

\({R^n}\)

Observe that for two relations \(R\) and \(S\)as follows:

\({M_{S^\circ R}} = {M_R} \odot {M_S}\)

So, for \(R = S\).

Then,\({M_{{R^2}}} = M_R^{(2)}\).

Proceed by induction as follows:

Suppose, \({M_{{R^{n - 1}}}} = M_R^{(n - 1)}\).

Write\({R^n}\) as \({R^{n - 1}}^\circ R\) and obtain as follows:

\(\begin{array}{c}{M_{Rn}} = {M_{{R^{n - 1}}^\circ R}}\\ = {M_R} \odot {M_{{R^{n - 1}}}}\\ = {M_R} \odot M_R^{(n - 1)}\\ = M_R^{(n)}\end{array}\)

Thus, \({\bf{M}}_R^{(n)}\) is the matrix represents the relation \({R^n}\).