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Let $S$ and $R$ be transitive relations on set $A$. Is $S∘R$ also transitive?

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    What are your thoughts? And how are the domains/codomains of $S$ and $R$ related? – 2017-01-13
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    Both are on the same set. It doesn't seem to be true for general case but I can't find the counterexample. – 2017-01-13

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$S=\{(2,3),(4,5)\}$ and $R=\{(1,2),(3,4)\}$ is a counterexample. They're both transitive, but $S\circ R=\{(1,3),(3,5)\}$ is not.

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    You can easily make $S$ and $R$ equivalence relations, with equivalence classes $\{\{1\},\{2,3\},\{4,5\}\}$ and $\{\{1,2\},\{3,4\},\{5\}\}$ respectively, by adding in more ordered pairs to each. The example would still work. – 2017-01-13
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    Thanks. My problem was I was thinking in terms of "smaller-than", equality or being a subset. – 2017-01-13
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    @dddwww This has a geometric interpretation of five points arranged in a staircase fashion, with $aSb$ meaning $a$ is in the same row as $b$ but in a higher column, and $aRb$ meaning $a$ is in the same column as $b$ but in a higher row. – 2017-01-13
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    Like this:$\begin{matrix}&&1\\&3&2\\5&4&\end{matrix}$ @dddwww – 2017-01-13