[Math] Quantified Proposition

discrete mathematicslogicpredicate-logic

I've been trying to translate the following sentences into quantified propositions by making sure I state all propositional functions that I use and any assumptions that I make.

There is exactly one person who hates everyone.

Let $H(x, y)$ be '$x$ hates $y$,'
where the domain of $x$ is all people in the world.

Then, $\exists x \forall y\ (\ H(x,y)\ \land \forall z\ (z \neq x) \rightarrow \neg H(z,y)\ ) $.

Can you see if I'm on the right track here?

Best Answer

Your statement, as it reads now, means:

There exists one such person $x$ that for every other person $y$, $x$ hates $y$ and nobody else hates $y$.

That is not what you want to state. For example, if

  • $A$ hates $A$, $B$ and $C$ and
  • $B$ hates $C$,

then the original statement (there exists a person who hates everybody) is true, but your statement is false (because it is not true that nobody else hates $C$)

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