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I am solving questions of a bachelors exam and was unable to solve this question and i am looking for help!!

Find Number of reflexive relations on the set {1,2,...,n}.

I know that R is called reflexive if for every a $\in $R a R a. But when I used it here 1 got that there would be only 1 reflexive relation ie each element goes to itself but that's wrong according to answers.

I don't have any idea on how too approach the question .

Waiting for your reply!

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  • $\begingroup$ Saying that $a\,R\,a$ for all $a$ doesn't preclude $a\, R \,b$ for some $b\neq a$. $\endgroup$ – lulu Nov 13 '20 at 15:43
  • $\begingroup$ Suppose the set is $\{1,2\}$ then $\{(1,1),(2,2),(1,2)\}$ is also reflexive, you are forgetting other elements. $\endgroup$ – PNDas Nov 13 '20 at 15:43
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As the set $S=\{1,2,\ldots, n\}$ has $n$ elements, the set of all pairs of elements in $S$ (i.e. $S\times S$) has $n^2$ elements.

Out of those, $n$ pairs are of the form $(x,x)$ for $x\in S$. Those pairs must belong to a reflexive relation $R$. As for the remaining $n^2-n$ pairs, each of them may or may not belong to the relation $R$, so the number of potential candidates for $R$ is $2^{n^2-n}$.

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But you can have whatever relation you want for $aRb$ with $a\neq b.$ Show that there are $2\binom{n}{2}$ of them. And either they are there or not. Giving a total of $2^{2\binom{n}{2}}.$

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