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If a set has $n$ elements, how many transitive relations are there on it?

For example if set $A$ has $2$ elements then how many transitive relations. I know the total number of relations is $16$ but how to find only the transitive relations? Is there a formula for finding this or is it a counting problem?

Also how to find this for any number of elements $n$?

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4 Answers 4

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There is no simple formula for this number (but see http://oeis.org/A006905 for the values for small $n$). The case $n=2$ is small enough that you can list out all 16 different relations and count the ones that are transitive. (You should get 13 of them.)

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  • $\begingroup$ It's easier to just find the three nontransitive relations on $\{a,b\};$; they are $\{(a,b),(b,a)\}\cup S$ where $S$ is a proper subset of $\{(a,a),(b,b)\}.$ $\endgroup$
    – bof
    Commented Feb 28, 2017 at 8:48
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    $\begingroup$ @bof: That's true for size $n = 2$ but not true for any sizes past that. E.g., for $n = 3$, only 33% are transitive; for $n = 4$, 6%; for $n = 5$, 0.5%; etc. $\endgroup$ Commented Nov 11, 2018 at 3:03
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Although there's no formula, results for small $n$ can be obtained by recursion. This paper proves that, if there are $T_n$ transitive relations and $P_n$ partial orders on an $n$-element set, and if we define $N_k\left( n\right):=\sum_{s=0}^k\binom{n}{s}S\left( n-s,\,k-s\right)$ where $S\left( n,\,k\right):=\frac{1}{k!}\sum_{i=1}^k\left( -1\right)^{k-i}\binom{n}{k}i^n$, then $T_n=\sum_{k=1}^n N_k\left( n\right)P_k$. Unfortunately, $P_n$ is also only known for small $n$; we can't obtain further $P_n$ with any known recursion, which in turn caps computing $T_n$.

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As noticed by @universalset, there are 13 transitive relations among a total of 16 relations on a set with cardinal 2. And here are they :)

enter image description here

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    $\begingroup$ Could you please explain why for instance the first one (on the top left corner) is transitive? I don't see it, should be at least need three element to define transitivity? $\endgroup$
    – user427132
    Commented Jul 9, 2018 at 17:08
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    $\begingroup$ You can test transitivity using the definition of transitivity and a the truth table for the implies operator (→) A relation is transitive if, in simple terms, a is related to b, and b is related to c implies that a is related to c. aRb & bRc → aRc The '→' works as follows: | p | q | p→q | | 0 | 0 | 1 | | 0 | 1 | 1 | | 1 | 0 | 0 | | 1 | 1 | 1 | So to answer Sam Farjamirad's question, In a two node graph with NO connections: aRb = FALSE bRc == FALSE aRb→bRc = TRUE So it is transitive $\endgroup$ Commented Jul 26, 2018 at 17:33
  • $\begingroup$ More briefly, transitivity is a "for all" statement, and when there are no relations, then it is vacuously true. $\endgroup$ Commented Nov 11, 2018 at 3:09
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Counting transitive relations on a set is probably very hard. Just recently, in this paper of mine titled, "On the number of transitive relations on a set", I was able to find several recursive relations and lower bounds for the number of transitive relations on a set.

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