$2$ Binary Relationships

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$($x$,$y$).$

$($iv$)$x$$y$($R$($x$,$y$)$R$($y$,$x$)).$

$($commutation$)($distribution$)($idempotence$)$

$($b$)$ Consider the natural numbers with the binary predicate R$($x$,$ y$)$ as $$x $$ y $=0.($i$)$ Check that the conditions $(1)$ and $(2)$ are true of R in this setting.

$($ii$)$ Translate conditions $(1)$ and $(2),$ when applied to this setting, into simple English sen