Q$4$ Guantifiers $[40]$ Suppose that the students at Memorial Uni$-$

versity 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 $S=JuP.$

We define class as a Boolean function taking two people as ar$-$

guments, which returns true if they are in a class together and

false if they are not. Also, the order of arguments does not matter:

AAx,yinS$*class(x,y)>$class$(y,x)$

$($a$)$ Express the following natural language sentences formally,

using quantifiers, sets, etc. $[10$ marks per part, $20$ total$]$

$($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.

$[10$ marks per part, $20$ total$]$

$($i$)$ AAxinJ$*EEyinJ*(x=y{)}^{{?}^{?}}$class$(x,y)$

$($ii$)$ EExinJ$*AAyinP*class(x,y)$