Answer
$a)$ $\exists x P(x)$
$b)$ $\neg \forall x P(x)$
$c)$ $(\forall x P(x)) \wedge(\forall y Q(y))$
$d)$ $\exists y \neg Q(y)$
$e)$ $\forall y Q(y)$
Work Step by Step
Given:
P(x)= “student $x$ knows calculus”.
Q(x)= “Class $y$ contains a student who knows calculus”.
$a)$
The given statement can be rewritten as ” there exists a student who knows calculus”.
Using the above interpretations we can then rewrite the statement as a mathematical expression:
$\exists x P(x)$
$b)$
The given statement can be rewritten as “Not all students know calculus”.
Using the above interpretation, we can then rewrite the statement as a mathematical expression:
$\neg \forall x P(x)$
$c)$
The given statement can be rewritten as” all students know Calculus and every class contains a student who knows calculus”.
Using the above interpretation, we can then rewrite the statement as a mathematical expression:
$(\forall x P(x)) \wedge(\forall y Q(y))$
$d)$
The given statement can be rewritten as “there exists a class that does not contain a student who knows calculus”
Using the above interpretations we can then rewrite the statement as a mathematical expression:
$\exists y \neg Q(y)$
$e) $
The given statement can be rewritten as “All classes contain a student who knows calculus”.
Using the above interpretations we can then rewrite the statement as a mathematical expression:
$\forall y Q(y)$
