I'm trying to understand how to negate a quantifier. I'm not that good at writing out a quantified logical statement either.
This is the statement: There is somebody that no-one is taller than. This is the negation: ( ∀ y) ( ∃ x) x is taller than y.
Can someone explain how they found the negation?
This is what I attempted:
Negation: Everybody is taller than somebody.
∀ x: everybody
∃ y: somebody
( ∀ x) ( ∃ y) x is taller than y.
I'm getting it flip flopped. I don't understand why.
Let's use $h(\;)$ as the height measure.
"There is somebody that no-one is taller than." $\equiv \overset{\tiny\text{some one}}{\exists y}~\overset{\tiny\text{no(t) one}}{\neg\exists x} ~\overset{\tiny\text{is taller than (the some one)}}{(h(x){>}h(y))}$
Negate it: $\neg\exists y~\neg\exists x ~(h(x)>h(y))~~\equiv ~~\forall y~\exists x~(h(x){>}h(y))$
That is "Everyone has someone that is taller than them", or "Everybody is shorter than somebody."