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# Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 3), (2, 4), (3, 1), (3, 5), (4, 3), (5, 1), (5, 2), and (5, 4). Find R^2 , R^3 , R^4 , R^5 , and R^6 .

Solution

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Step 1
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Given:

\begin{align*} A&=\{1,2,3,4,5\} \\ R&=\{(1,3),(2,4),(3,1),(3,5),(4,3),(5,1),(5,2),(5,4)\} \end{align*}

Let $\textbf{M}$ be a -matrix that represent the relation $R$.

Since the set $A$ contains 5 elements, we represent the relationship $R$ by a $5\times 5$ matrix $\textbf{M}$.

If $(i,j)\in R$, then the element in the $i$th row and in the $j$th column of the matrix is 1, else the element is 0.

$\textbf{M}=\begin{bmatrix}0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0\\ 1 & 1 & 0 & 1 & 0\end{bmatrix}$

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