How does negating work in quantified statements What is the negation of “all users are online”? **p(x) represents users online** Answer: Not all users are online, i.e. at least one user is offline. Symbolically: ¬(∀x p(x)) ≡ ∃x ¬p(x). **Hi I just wanted to understand how the negation of all users are online** becomes ¬(∀x p(x)) ≡ ∃x ¬p(x). **I thought If I negated the statement all users are online I would get just** ∃x ¬p(x). **I really don't understand how their is a equivalence symbol in the negation**

The $\equiv$ is not meant to be part of the symbolization.

Rather, the provided answer is indicating that $\
eg \forall x \ P(x)$ is logically equivalent to $\exists x \ \
eg P(x)$

That is, if you start with negating the statement that 'all users are online', you get $\
eg \forall x \ P(x)$ as an answer, but if they don't want the answer to start with a negation and have any quantifiers at the start, you can use the equivalence to change it into $\exists x \ \
eg P(x)$