How do you negate a sentence with existential and universal quantifiers?

Question

I am having a trouble some time understanding the negation of ∀.

For example:

Let $F$ be a set of food critics and let $R$ be a set of restaurants. The statement $V(f,r)$ means "a food critic $f$ ate at restaurant $r$" $∃f$ $∈$ $S$ and $∃c$ $∈$ $S$ such that $f$ and $c $ are not the same and $∀r$ $∈$ $R$, $V(f, r) → V(c, r)$.

Would the negation be: If some food critics $f$ ate at restaurant $r$ then some food critics $c$ ate at restaurant $r$.

Or If one food critic $f$ are at restaurant $r$ then another food critic $c$ ate at $r$?

I hope this makes sense thanks in advance

Answer 1

For every $f$ and for every $c$, with $f \ne c$, there exists an $r$, such that $f$ ate at $r$ but $c$ did not. The rule is you change every existential quantifier to universal, every universal to existential and finally negate the assertion at the end. The negation of "$V(f, r) \Rightarrow V(c, r)$" is "$V(f, r)$ and not $V(c, r)$".

In case the $f\ne c$ bothers you, write the original statement as "there exists a pair $(f, c)$ with $f \ne c$ such that...". Then the negation is "for every pair $(f, c)$ with $f \ne c$, ...$.

Answer 2

You can do it mechanically if you write the statement in proper notation. Your fourth paragraph would be translated as $$\exists f \in S \left(\exists c \in S (f\neq c \wedge \forall r V(f,r) \implies V(c,r))\right)$$ If you want to negate this you just follow the rules $$\lnot \exists f \in S \left(\exists c \in S (f\neq c \wedge \forall r V(f,r) \implies V(c,r)\right) \iff\\ \forall f\in S\lnot \left(\exists c \in S (f\neq c \wedge \forall r V(f,r) \implies V(c,r)\right) \iff\\ \forall f\in S\forall c \in S \lnot(f\neq c \wedge \forall r V(f,r) \implies V(c,r))\iff \\ \forall f\in S\forall c \in S (f= c \vee \lnot\forall r V(f,r) \implies V(c,r))\iff \\ \forall f\in S\forall c \in S (f= c \vee \exists r V(f,r)\land \lnot V(c,r)) $$