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One possible relation ,

$\begin{align*} &R = \left \{{\color{blue}{\left ( 1,1 \right )}},{\color{red}{\left ( 1,3 \right )}},{\color{blue}{\left ( 2,2 \right )}},{\color{red}{\left ( 3,1 \right )}} \right \} \\ \end{align*}$

In this example two types of pairs are used.

  • Self-loop pairs (blue ones)
  • Other pairs (red ones)

 Here is the matrix representation of such relation.

  1. Self-loops pairs selected out of $7$ gray places.
  2. Other pairs selected out of $\begin{align*} \frac{7^2-7}{2} = 21 \end{align*}$ green places. 
  3. Other pairs will only be selected in even numbers. Like $\{1,2\},\{2,1\}$ Here two pairs.
  4. To select two Other pairs we can choose only one green place out of $21$.
  5. To select four Other pairs we can choose only two green places out of $21$ and so on..

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