Let $R$ be a relation on a set $X$. Then prove that $R\cup R^{-1}$ is the smallest symmetric relation containing $R$ and $R\cap R^{-1}$ is the largest symmetric relation contained in $R$.

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    $\begingroup$ Do you define $R^{-1}$ to be $R^{-1}=\{(x,y)\in X\times X:(y,x)\in R\}$? $\endgroup$ – Floris Claassens May 3 at 10:38
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    $\begingroup$ Welcome to MathSE. Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering. $\endgroup$ – N. F. Taussig May 3 at 10:42
  • $\begingroup$ This is fairly straightforward verification of the definitions of "symmetric relation" and "contained/containing". It would be helpful for us to help you if you can at least point out the problems you are having. $\endgroup$ – Asaf Karagila May 3 at 14:18

The largest symmetric relation containing R is X×X.
If R = {(0,1)}: R$^{-1}$ = {(1,0)}; R $\cap$ R$^{-1}$ is empty; does not contain R.

On the other hand if R is a relation, then Rs = R $\cup$ R$^{-1}$ is symmetric and contains R.
There are two cases to prove Rs is symmetric. What are they?
To prove Rs is the smallest symmetric relation containing R,
assume Q is a symmetric relation containg R.
Prove Rs is a subset of Q.


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