I'm struggling to understand this concept entirely.
I'm supposed to apply the equivalence law being asked to the left side of the formula, but I'm totally lost. Can anyone guide me? I've been trying to understand this concept for a few days now, reading all the text in the book, going over class notes, reading forums, etc, but its just not coming to me.
A = set of Americans. L(x, y) = “x loves y”. Use Quantifier negation laws with bounded quantifiers:
∃x ∈ A ¬∀yL(x,y) ⇔ ∃x ∈ A ∃y¬L(x,y) (??)
The negation of a quantifier is given by: $$ \neg \forall x P(x) \iff \exists x \neg P(x) $$ $$ \neg \exists x P(x) \iff \forall x \neg P(x) $$ you can easily convince by yourself that this is true, or see here.
So, $\neg \forall y L(x,y)$ becomes $\exists y \neg L(x,y)$
and this simply means that: there exists an American that does not love all americans, is equivalent to: there exists an American such that there exist another american (at least) that he does not love.
In other words, to say that there is an American who does not love all Americans, does not means that he hates them all, but that does not love at least one.