What does this formula state?

Question

I have the following formula in predicate logic:

∀x∀y∃z(R(x, z) ∧ ¬R(y, z))

I am not entirely sure what it states as for me it states a contradiction. Asuming it is a graph and Relation R(x,y) means that x is adjacent to y, this formula says that for all nodes in the graph they are adjacent to z and they aren't adjacent to z, or do I have it wrong and one should make the distinguishment between x and y, meaning that for a some nodes x in the graph they are adjacent to z and for other nodes y they aren't adjacent to z?

Thank youu:)

Answer 1

Generally $$(A \land \neg B) = \neg(\neg A \lor B)= \neg (A \Rightarrow B)$$ So you have $$\forall x \forall y \exists z \left( \neg(R(x,z) \Rightarrow R(y,z)) \right)$$

Answer 2

The formula states that for any $x$ and $y$ there is a $z$ that is related to $x$ but not $y$. Since the quantifiers $\forall x\forall y$ do not necessarily imply $x\neq y$, you are correct that this formula can never be true. In order to obtain a formula that could be true, there needs to be an assumption that $x\neq y$. Something like: $$ \forall x\forall y (x\neq y\rightarrow \exists z(R(x,z) \wedge \neg R(y,z))) $$