Here are two ways of looking at the divisibility relation, a dyadic relation of fundamental importance in number theory.
Table 1 shows the first few ordered pairs of the relation on positive integers that corresponds to the relative term, “divisor of”. Thus, the ordered pair 
![\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 1. Elementary Relatives for the ``Divisor Of" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1\!\!:\!\!1&1\!:\!2&1\!:\!3&1\!:\!4&1\!:\!5&1\!:\!6&1\!:\!7&1\!:\!8&1\!:\!9&1\!:\!10&\dots \\ 2&&2\!:\!2&&2\!:\!4&&2\!:\!6&&2\!:\!8&&2\!:\!10&\dots \\ 3&&&3\!:\!3&&&3\!:\!6&&&3\!:\!9&&\dots \\ 4&&&&4\!:\!4&&&&4\!:\!8&&&\dots \\ 5&&&&&5\!:\!5&&&&&5\!:\!10&\dots \\ 6&&&&&&6\!:\!6&&&&&\dots \\ 7&&&&&&&7\!:\!7&&&&\dots \\ 8&&&&&&&&8\!:\!8&&&\dots \\ 9&&&&&&&&&9\!:\!9&&\dots \\ 10&&&&&&&&&&10\!:\!10&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%7Cc%7C%7C%2A%7B11%7D%7Bc%7D%7C%7D++%5Cmulticolumn%7B12%7D%7Bc%7D%7B%5Ctext%7BTable+1.+Elementary+Relatives+for+the+%60%60Divisor+Of%22+Relation%7D%7D+%5C%5C%5B4pt%5D++%5Chline++i%7Cj+%261%262%263%264%265%266%267%268%269%2610%26%5Cldots+%5C%5C++%5Chline%5Chline++1%261%5C%21%5C%21%3A%5C%21%5C%211%261%5C%21%3A%5C%212%261%5C%21%3A%5C%213%261%5C%21%3A%5C%214%261%5C%21%3A%5C%215%261%5C%21%3A%5C%216%261%5C%21%3A%5C%217%261%5C%21%3A%5C%218%261%5C%21%3A%5C%219%261%5C%21%3A%5C%2110%26%5Cdots+%5C%5C++2%26%262%5C%21%3A%5C%212%26%262%5C%21%3A%5C%214%26%262%5C%21%3A%5C%216%26%262%5C%21%3A%5C%218%26%262%5C%21%3A%5C%2110%26%5Cdots+%5C%5C++3%26%26%263%5C%21%3A%5C%213%26%26%263%5C%21%3A%5C%216%26%26%263%5C%21%3A%5C%219%26%26%5Cdots+%5C%5C++4%26%26%26%264%5C%21%3A%5C%214%26%26%26%264%5C%21%3A%5C%218%26%26%26%5Cdots+%5C%5C++5%26%26%26%26%265%5C%21%3A%5C%215%26%26%26%26%265%5C%21%3A%5C%2110%26%5Cdots+%5C%5C++6%26%26%26%26%26%266%5C%21%3A%5C%216%26%26%26%26%26%5Cdots+%5C%5C++7%26%26%26%26%26%26%267%5C%21%3A%5C%217%26%26%26%26%5Cdots+%5C%5C++8%26%26%26%26%26%26%26%268%5C%21%3A%5C%218%26%26%26%5Cdots+%5C%5C++9%26%26%26%26%26%26%26%26%269%5C%21%3A%5C%219%26%26%5Cdots+%5C%5C++10%26%26%26%26%26%26%26%26%26%2610%5C%21%3A%5C%2110%26%5Cdots+%5C%5C++%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26++%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots+%5C%5C++%5Chline++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 1. Elementary Relatives for the ``Divisor Of\" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1\!\!:\!\!1&1\!:\!2&1\!:\!3&1\!:\!4&1\!:\!5&1\!:\!6&1\!:\!7&1\!:\!8&1\!:\!9&1\!:\!10&\dots \\ 2&&2\!:\!2&&2\!:\!4&&2\!:\!6&&2\!:\!8&&2\!:\!10&\dots \\ 3&&&3\!:\!3&&&3\!:\!6&&&3\!:\!9&&\dots \\ 4&&&&4\!:\!4&&&&4\!:\!8&&&\dots \\ 5&&&&&5\!:\!5&&&&&5\!:\!10&\dots \\ 6&&&&&&6\!:\!6&&&&&\dots \\ 7&&&&&&&7\!:\!7&&&&\dots \\ 8&&&&&&&&8\!:\!8&&&\dots \\ 9&&&&&&&&&9\!:\!9&&\dots \\ 10&&&&&&&&&&10\!:\!10&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}")
Table 2 shows the same information in the form of a logical matrix. This has a coefficient of 
![\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 2. Logical Matrix for the ``Divisor Of" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1&1&1&1&1&1&1&1&1&1&\dots \\ 2& &1& &1& &1& &1& &1&\dots \\ 3& & &1& & &1& & &1& &\dots \\ 4& & & &1& & & &1& & &\dots \\ 5& & & & &1& & & & &1&\dots \\ 6& & & & & &1& & & & &\dots \\ 7& & & & & & &1& & & &\dots \\ 8& & & & & & & &1& & &\dots \\ 9& & & & & & & & &1& &\dots \\ 10&& & & & & & & & &1&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7B%7Cc%7C%7C%2A%7B11%7D%7Bc%7D%7C%7D++%5Cmulticolumn%7B12%7D%7Bc%7D%7B%5Ctext%7BTable+2.+Logical+Matrix+for+the+%60%60Divisor+Of%22+Relation%7D%7D+%5C%5C%5B4pt%5D++%5Chline++i%7Cj+%261%262%263%264%265%266%267%268%269%2610%26%5Cldots+%5C%5C++%5Chline%5Chline++1%261%261%261%261%261%261%261%261%261%261%26%5Cdots+%5C%5C++2%26+%261%26+%261%26+%261%26+%261%26+%261%26%5Cdots+%5C%5C++3%26+%26+%261%26+%26+%261%26+%26+%261%26+%26%5Cdots+%5C%5C++4%26+%26+%26+%261%26+%26+%26+%261%26+%26+%26%5Cdots+%5C%5C++5%26+%26+%26+%26+%261%26+%26+%26+%26+%261%26%5Cdots+%5C%5C++6%26+%26+%26+%26+%26+%261%26+%26+%26+%26+%26%5Cdots+%5C%5C++7%26+%26+%26+%26+%26+%26+%261%26+%26+%26+%26%5Cdots+%5C%5C++8%26+%26+%26+%26+%26+%26+%26+%261%26+%26+%26%5Cdots+%5C%5C++9%26+%26+%26+%26+%26+%26+%26+%26+%261%26+%26%5Cdots+%5C%5C++10%26%26+%26+%26+%26+%26+%26+%26+%26+%261%26%5Cdots+%5C%5C++%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26++%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots%26%5Cldots+%5C%5C++%5Chline++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 2. Logical Matrix for the ``Divisor Of\" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1&1&1&1&1&1&1&1&1&1&\dots \\ 2& &1& &1& &1& &1& &1&\dots \\ 3& & &1& & &1& & &1& &\dots \\ 4& & & &1& & & &1& & &\dots \\ 5& & & & &1& & & & &1&\dots \\ 6& & & & & &1& & & & &\dots \\ 7& & & & & & &1& & & &\dots \\ 8& & & & & & & &1& & &\dots \\ 9& & & & & & & & &1& &\dots \\ 10&& & & & & & & & &1&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}")
Just as matrices in linear algebra represent linear transformations, these logical arrays and matrices represent logical transformations.
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