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> Let $R$ be a reflexive and transitive relation on $X$. Show that $R \cap R^{−1}$ is an equivalence relation on $X$.
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I am stuck and feel like I am doing something wrong, any help would be much appreciated!

Here is what I have so far:

$R = \{1,2,3\}$

$X = \{2,3,4,5,6\}$

$R$ on $X = \{(1,2), (1,3),(1,4),(1,5),(1,6),(2,4),(2,6),(3,6)\}$

$R^{-1} = \{1,1/2,1/3\}$

$R \cap R^{-1} = \{1\}$

$R^{-1}$ on $X = \{(1,2),(1,3),(1,4),(1,5)(1,6)\}$

TRUE

Am I completely off-base or am I close?

Thanks!