Use quantifiers to express and then negate

Question

There is a woman who has taken a flight in every airline in the world. I used "For every x(woman), P(x)" where P(x) : taken flight in every airline. Is it better than using 3 variables?

Answer 1

"There is a woman" implies an existential quantifier.

Also, having quantifiers both outside and inside $P$ makes it hard to tell what you are trying to achieve.

The most natural way to me, would be to make $W$ to be the set of women, $A$ the set of airlines, and $P(w,a)=$"$w$ flew in airline $a$". Then your statement is $$ \exists w\in W,\ \forall a\in A,\ P(w,a). $$

Answer 2

As Martin says, you need an existential quantifier for the woman, and a universal for the airlines. But given that it says 'a flight', I would also introduce an existential quantifier for those flights.

So, let's use predicates:

$W(x): x$ is a woman

$A(x) : x$ is an airline

$F(x) : x$ is a flight

$O(x,y) : x$ is on $y$

$B(x,y) : x$ belongs to $y$

Then the sentence becomes:

$$\exists x (W(x) \land \forall y (A(y) \rightarrow \exists z (F(z) \land B(z,y) \land O(x,z))))$$

Answer 3

Statemrnt. Some woman w who for all airlines a, w fly on a.
Negation. For all women w, some airline a with w not fly on a.