Question $3$: This exercise uses the function $\backslash (\backslash$operatorname$\{$MapColor$\}($x$)\backslash )$ and predicates $\backslash (\backslash$operatorname$\{$In$\}($x$,$ y$)\backslash ),$ Borders$(\backslash ($ x$,$ y $\backslash )),$ and Country$(\backslash ($ x $\backslash )),$ whose arguments are geographical regions, along with constant symbols for various regions. In each of the following we give an English sentence and a number of candidate logical expressions. For each of the following expressions, state whether it $-$

a$)$ correctly expresses the English sentence;

b$)$ is syntactically invalid and therefore meaningless; or

c$)$ is syntactically valid but does not express the meaning of the English sentence.

All countries that border Ecuador are in South America.

A$.\backslash (\backslash$forall $\backslash$mathrm$\{$c$\}\backslash )$ Country$($c$)\backslash (\backslash$wedge $\backslash )$ Border $($c$,$ Ecuador$)\backslash (\backslash$Rightarrow $\backslash$operatorname$\{$In$\}(\backslash$mathrm$\{$c$\}\backslash ),$ SouthAmerica$)$

B$.\backslash (\backslash$forall $\backslash )$ c Country$($c$)\backslash (\backslash$Rightarrow$[\backslash )$ Border $\backslash ((\backslash$mathrm$\{$c$\}\backslash ),$ Ecuador $\backslash ()\backslash$Rightarrow $\backslash$operatorname$\{$In$\}(\backslash$mathrm$\{$c$\}\backslash ),$ SouthAmerica $\backslash ()]\backslash )$

There is a country that borders both Iraq and Pakistan.

C$.\backslash (\backslash$exists $\backslash$mathrm$\{$c$\}\backslash )$ Country$($c$)\backslash (\backslash$wedge $\backslash )$ Border $($c$,$ Iraq$)\backslash (\backslash$wedge $\backslash )$ Border $($c$,$ Pakistan$).$

D$.\backslash (\backslash$exists $\backslash$mathrm$\{$c$\}\backslash )$ Country $\backslash ((\backslash$mathrm$\{$c$\})\backslash$Rightarrow$[\backslash )$ Border $\backslash ((\backslash$mathrm$\{$c$\}\backslash ),$ Iraq$)\backslash (\backslash$wedge $\backslash )$ Border $\backslash ((\backslash$mathrm$\{$c$\}\backslash ),$ Pakistan $\backslash ()]\backslash ).$