Negating a Universal Quantification

The statement

‘ Every student in your class has taken a course in calculus ’

\forall x P (x)

where $P (x)$ is the statement $‘ x has taken a course in calculus ’$ and the domain consists of all the students in you class
What is the negation?

\begin{aligned} ‘ & It is not the case that \\ every student in your class has taken a course in calculus ’ \end{aligned}

How can this also be expressed?

‘ There is a student in your class who has not taken a course in calculus ’

Therefore...

\neg \forall x P (x) \equiv \exists x \neg P (x)