0

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.

Fernando Martinez
  • 6,698
  • 19
  • 74
  • 108

1 Answers1

0

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 $y$.

Badoe
  • 481
  • 1
    The use of a set and membership is totally optional for a single-sorted theory (and even a many-sorted theory can usually avoid it). Personally, I don't think it adds much, pedagogically. – Malice Vidrine Jun 11 '14 at 20:32
  • @MaliceVidrine I agree. But it's confusing in the original post. Fernando used $x$ to represent all people and then used $\exists x$ in the quantifier to represent some person. – Badoe Jun 11 '14 at 20:36
  • D'oh. Somehow I had totally missed that; in that case it sure does represent an improvement. (For the OP's benefit, it would also be well understood verbiage to say something like "let all variables range over the set of all people/yurts/people in yurts".) – Malice Vidrine Jun 11 '14 at 21:01
  • I see thanks this makes sense – Fernando Martinez Jun 12 '14 at 15:45