In my assignment, I have the following question: Find a model with the domain {a, b, c, d} so that:
M |= ∀x∃yT(x, y),
M |= ¬∃x∀yT(x, y),
M |= ¬∃y∀xT(x, y),
and T^M is a transitive relation.
What I struggle to understand is how the different quantifiers and variables affect eachother. I believe the first rule states that for ALL x, there is an Y. However, THERE ISN'T an X for every Y, and there isn't an Y for every X.
Doesn't the first contradict with the two latter?
You have understood the first.
The second says you can't have all of $(a,a), (a,b), (a,c), (a,d)$ and similar sets.
The third says you can't have all of $(a,a), (b,a), (c,a), (d,a)$ and similar sets.
Once you insist that $T$ be an equivalence relation these are redundant.