Image of an exam question I am revising link: 
For (i) I have stated the relation is reflexive as $\forall x ∈ \Bbb R, xPx$ is reflexive as $x^3 \ge x $
For (ii) I have stated that the relation is not symmetric as $\forall x,y ∈ \Bbb R, xRy$ does not equal $yRx$, as for the values $x=2, y=1, x^3 - y \ge y^3 - x$ does not equal to $y^3 - x \ge x^3 - y$
For (iii) I have stated that the relation is transitive as as $xPy$ and $yPz$ imply $xPz \forall x,y,z ∈ P,$ as $\forall x,y,z$ that are elements of $R$ and that satisfy $P$, $(a,b)$ $a$ must always be greater than $b$, therefore $\forall x,y ∈ P x\ge y, \forall y,z ∈ P y\ge z,$ therefore $x\ge z$
The last one is a tad all over the place, but I just wanted to know if I am on the right track and if my answers are correct, if not I would love to know why so I can improve myself and see where I went wrong as to not make the same mistake in the future. Any and all help would be greatly appreciated.