A = (∀x∀y∀z(P xy → (P yz → P xz)))
B = (∀x∀y(P xy → (P yx → x = y)))
C = (∀x∃yP xy) → (∃y∀xP xy)
I want to show that {A, B} does not imply C by constructing a structure.
What I've done so far is just translated it to English so I can visualize it better:
A: For all x, y, z: If XY, then YZ implies XZ
B: For all x, y: If XY, then YX implies X=Y
C: IF For all X there exists some Y where XY, THEN there exists a Y such that for all X, XY.
I'm going about this problem by using C as a constraint, since it's what we want to contradict. But this is where I'm stuck. I would like to know how to actually construct the required structure.
Hint: A says that $P$ is a transitive relation. B says that $P$ is anti-symmetric. This is perhaps obscured by the sentences using only implication "$\to$". For example, A is usually stated in the equivalent form $\forall x \forall y \forall z\,((P(x,y) \land P(y,z)) \to P(x,z))$.
For any set of numbers, finite or infinite, the usual ordering $\le$ is an example of such a relation. (Actually, so is $<$, but B is true for $<$ "vacuously", as there are no numbers $x,y$ such that $x<y$ and $y<x$.) If you interpret $P$ as $\le$, consider what C says about $\le$. The antecedent (first part of the implication) is true; how about the consequent (the second part)?