Translate the following predicate logic into English sentences.

$($I$)$x$$y$((($x $<0)($y $<0))->($xy $>0))$

$($II$)$x$$y$(($x $=0)($y $=0)($xy $=0))$

$($III$)$ U$($x$)$ : Country x is part of the United Nations

P$($x$,$ y$)$ : Countries x and y have signed a peace treaty with each other

Domain of x and y: All countries in the world.

$($i$)$x$$y$($U$($x$)->$ P$($x$,$ y$))$

$($ii$)$x$$y$($U$($x$)->($P$($x$,$ y$)))$

$($iii$)$x$$y$$z$(($U$($x$)$U$($y$)($x $=$ y$))->($P$($x$,$ z$)$ P$($y$,$ z$)))$