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How many reflexive relations are there on a set of $n$ elements? I did the problem and I got the answer $2 ^ {n ^ 2}$. Is it correct? Thanks for the help..!!

yoknapatawpha
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user126408
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    I think that your formula fails for example, when $n=1$. I can only think of a single reflexive relation on a singleton set. Try again! Hint: the number of independent binary choices that you can make is less than $n^2$. – Jyrki Lahtonen Feb 10 '14 at 21:52
  • The other answer I got is 2 ^ n. Is it correct? – user126408 Feb 10 '14 at 22:00
  • Does that pass the lithmus test I described in my first comment? Think! Don't just toss formulas around. – Jyrki Lahtonen Feb 10 '14 at 22:02

2 Answers2

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Hint: Note that if $X$ is a set with $n$ elements, then there are $n^2-n$ elements in $X\times X$ that are not of the form $\langle x,x\rangle$ for some $x\in X.$ Any, all, or none of these $n^2-n$ elements can be members of a reflexive relation on $X$. (Why?) What about the other $n$ elements of $X\times X$?

Cameron Buie
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$2^{n^2}$ is find for how many number of relations on a set on n elements. $2^{n(n-1)}$ is find for how many number of reflexive relations on set with $n$ elements.

drhab
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Salman
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