Question

To calculate

a) The matrix representing \({R_1} \cup {R_2}\).

b) The matrix representing \({R_1} \cap {R_2}\).

c) The matrix representing the composite relation \({R_2}^\circ \;{R_1}\).

d) The matrix representing the composite relation.

e) The matrix representing the direct sum of the relations \({R_1}\) and \({R_2}\).

Short answer

a) The matrix is \({M_R} = \left( {\begin{array}{*{20}{l}}0&1&0\\1&1&1\\1&1&1\end{array}} \right)\).

b) The matrix is \({M_R} = \left( {\begin{array}{*{20}{l}}0&1&0\\0&1&0\\1&0&0\end{array}} \right)\).

c) The matrix is \({M_R} = \left( {\begin{array}{*{20}{l}}1&1&0\\1&1&0\\1&1&0\end{array}} \right)\).

d) The matrix is \({M_R} = \left( {\begin{array}{*{20}{l}}0&1&1\\1&1&1\\0&1&0\end{array}} \right)\).

e) The matrix is \({M_R} = \left( {\begin{array}{*{20}{l}}0&1&0\\1&0&1\\0&1&1\end{array}} \right)\).

Step-by-step solution

Given data 

The matrices \({M_{{R_1}}}\) and \({M_{{R_2}}}\) of the relations \({R_1}\) and \({R_2}\) respectively.

Concept of Matrix

The ordered pair\((i,j)\)belongs to the relation if and only if the\({(i,j)^{th}}\)entry in the matrix is\(1\).

Calculate the matix \({R_1} \cup {R_2}\)

a)

The matrix of the union is given as:

\(\begin{array}{l}{M_R} = {M_{{R_1}}} \vee {M_{{R_2}}}\\{M_R} = \left( {\begin{array}{*{20}{l}}0&1&0\\1&1&1\\1&1&1\end{array}} \right)\end{array}\)

Calculate the matix \({R_1} \cap {R_2}\) 

b)

The matrix of the intersection is given as:

\(\begin{array}{l}{M_R} = {M_{{R_1}}} \wedge {M_{{R_2}}}\\{M_R} = \left( {\begin{array}{*{20}{l}}0&1&0\\0&1&0\\1&0&0\end{array}} \right)\end{array}\)

Calculate the matrix \({R_2}^\circ \;{R_1}\) 

c)

The matrix of the composite is given as:

\(\begin{array}{l}{M_R} = {M_{{R_2}}}^\circ \;{M_{{R_1}}}\\{M_R} = \left( {\begin{array}{*{20}{l}}1&1&0\\1&1&0\\1&1&0\end{array}} \right)\end{array}\)

Calculate the composite matrix 

d)

The matrix of the composite is given as:

\(\begin{array}{l}{M_R} = {M_{{R_1}}}^\circ \;{M_{{R_2}}}\\{M_R} = \left( {\begin{array}{*{20}{l}}0&1&1\\1&1&1\\0&1&0\end{array}} \right)\end{array}\)

Calculate the sum of matrix \({R_1}\) and \({R_2}\)

e)

The matrix of the sum of the relation is given as:

\(\begin{array}{l}{M_R} = {M_{{R_1}}} \oplus \;{M_{{R_2}}}\\{M_R} = \left( {\begin{array}{*{20}{l}}0&1&0\\1&0&1\\0&1&1\end{array}} \right)\end{array}\)