Suppose that the students at Memorial University all live in either St$.$ Johns or Mount Pearl. Let J be the set of students in St$.$ Johns, and P be the set of students in Mount Pearl. Then the set of all students S $=$ J $\backslash$cup P $.$We define class as a Boolean function taking two people as arguments, which returns true if they are in a class together and false if they are not. Also, the order of arguments does not matter:

$$x$,$ y $$ S $$ class$($x$,$ y$)$ class$($y$,$ x$)$

$($a$)$ Express the following natural language sentences formally, using quantifiers, sets, etc.

$($i$)$ Every student from Mount Pearl is in a class with somebody from St$.$ Johns.

$($ii$)$ There is at least one student from St$.$ Johns who is not in any classes with a student from Mount Pearl

$($b$)$ Express the following formal sentences in clear English.

$($i$)$x $$ J $$y $$ J $($x $=$ y$)$ class$($x$,$ y$)$ii$)$x $$ J $$y $$ P $$ class$($x$,$ y