proof or deproof linear equivalence of X, X is an amount.

Question

Again I am stuck at some proof. I need to proof or deproof that for all linear equivalences:

R:(X,X) is R =

So far I think it is correct because we get symmetry and linearity, but I have troubles to proof it. Any help is upvoted immediately.

here the def of linearity:

symmetry here:

transitiv:

reflexiv:

linear equivalent is: transitiv, reflexiv, symmetric and linear

Answer 1

It seem to be true.

You use symmetry and linearity according to the second formulations. Since $R$ is symmetric you have $R^{-1}\subseteq R$ and therefore $R = R^{-1}\cup R$ which by linearity will be equal to $\nabla_{X,X}$