How would I do the following quantifier and their negation
No one loves everybody.
or could you say : everybody does not love someone?
x is all people
So in symbolic this would be $\forall x, \exists y,$ x does not love y.
and the denial is
Someone loves everyone.
$\exists x,\forall y$, x loves y
My second ?
Everybody loves someone
$\forall x, \exists y$, x loves y
the denial
someone does not love everyone.
$\exists x, \forall y$, x does not love everyone.
The first is correct. But the notation can be improved. Usually we use an uppercase letter to represent a set and a lower case letter to represent an element in the set.
So a better way to state is "Let $X$ be all people." Then your first statement is:
$\forall x \in X, \exists y \in X$, such that $x$ does not love $y$.
Similar for other statements.
However, the denial for the second statement is a little off. It should be:
someone does not love anyone
or equivalently
someone loves no one.
The quantifier for the denial would be
$\exists x \in X, \forall y \in X$, $x$ does not love $y$.