$5.$ Consider the following interpretations, presented diagrammatically, and the given sentences. $($a$)$ Consider the interpretation in this Euler diagram and evaluate the following sentences. For each sentence, say whether it is true or false in the interpretation, and briefly explain your reasoning. i$.$ x$($B x $->$ C x$)$; ii$.$ x$($C x $($A x $$ D x$))$; iii. $$ x$($D x $->($A x $->($C x $$ B x$)))$; iv$.$ x$($B x $->($ D x $$ C x$)).($b$)$ Consider the interpretation in this directed graph for the twoplace predicate $\text{'}$ R $\text{'}$ and evaluate the following sentences. For each sentence, say whether it is true or false in the interpretation, and briefly explain your reasoning. $($Note: the node a is named $\text{'}$ a $\text{'},$ c is named $\text{'}$ c $\text{'},$ etc.$)$ i$.$ x$($R x C$)$; ii$.$ x $$ y$($R y x $$ x $!=$ y$)$; iii. $$ x $$ y$(($R x y $$ R y x$)->$ x $!=$ c$)$; iv$.$ x$($ y$($R x y $->$ x$=$y$)$ z$($ w$($R z w $->$ w$=$z$)->$ z$=$x$)).$