Let R_1 and R_2 be a relations on a set

Lloyd Allen 2021-11-19 Answered
Let \(\displaystyle{R}_{{1}}\) and \(\displaystyle{R}_{{2}}\) be a relations on a set A represented by the matrices below:
\[M_{R1}=\begin{bmatrix}0&1&0\\1&1&1\\1&0&0\end{bmatrix}\]
\[M_{R2}=\begin{bmatrix}0&1&0\\0&1&1\\1&1&1\end{bmatrix}\]
Find the matrix that represents \(\displaystyle{R}_{{1}}\cap{R}_{{2}}\)

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Expert Answer

romancemrf
Answered 2021-11-20 Author has 606 answers

Consider the given matrices, \[M_{R1}=\begin{bmatrix}0&1&0\\1&1&1\\1&0&0\end{bmatrix}\ and\ M_{R2}=\begin{bmatrix}0&1&0\\0&1&1\\1&1&1\end{bmatrix}\]
Note that, the intersection \(\displaystyle{a}\cap{B}\) is all elements that are both in A and in B.
The matrix corresponding to the intersection of two relations is the meet of the matrices representing each of the relations.
\(\displaystyle{M}_{{{R}_{{1}}\cap\ {R}_{{2}}}}={M}_{{{R}_{{1}}}}\cap{M}_{{{R}_{{2}}}}\)
\[=\begin{bmatrix}0\wedge0&1\wedge1&0\wedge0\\1\wedge0&1\wedge1&1\wedge1\\1\wedge1&0\wedge1&0\wedge1\end{bmatrix}\]
\[=\begin{bmatrix}0&1&1\\0&1&1\\1&0&0\end{bmatrix}\]
Thus, the matrix that represents \[R_1\cap R_2\ is\ =\begin{bmatrix}0&1&1\\0&1&1\\1&0&0\end{bmatrix}\]

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