ips

Let R be a binary predicate such that the following are true.

$(1)$x$$y

$($R$($x$,$ y$)=$ R$($y$,$ x$)$

$.$

$(2)$x$$yR$($x$,$ y$).$

$($a$)$ Prove or disprove whether each of the following are logically implied by conditions $(1)$ and $(2).$

$($i$)$x$$yR$($x$,$ y$).$

$($ii$)$xR$($x$,$ x$).$

$($iii$)$y$$xR