Understanding interpretation First-Order Logic

Question

I'm currently learning about interpretations in first-order logic and I found some slides online about the subject which were very helpful. There's an exercise in these slides with answers, but I'm having trouble understanding what makes them correct. For convenience, the slides can be found at: http://iaoa.org/isc2014/uploads/SlidesWassermann02.pdf

The exercise in question is this:

For each of the following sentences, give a logical interpretation that makes that sentence false and the other two sentences true:
(a) ∀x∀y∀z((P(x, y) ∧ P(y, z)) → P(x, z))
(b) ∀x∀y((P(x, y) ∧ P(y, x)) → x = y)
(c) ∀x∀y(P(a, y) → P(x, b))

Instead of creating a wall of text, I'm just going to post the interpretation which makes sentence (a) false and sentences (b) and (c) true:

D = {d1, d2, d3}
I[P] = {(d1, d2),(d2, d3)}
I[a] = d3
I[b] = d2

So here's my question: What is it in this interpretation that makes sentence a false, and the others true? I've already read a lot about interpretations online, and even understood simpler examples, but this one seem to be a bit more advanced than those. I just need to understand the meaning behind the answer. In short: Why isn't sentence (a) true and why aren't sentence (b) and (c) false in this interpretation?

Answer 1

Why is the sentence $∀x∀y∀z((P(x, y) ∧ P(y, z)) → P(x, z))$ [call it : (A)] false in the said interpretation ?

Because in the said interpretation, with domain $D$, we have that the (binary) predicate letter $P$ is interpreted with the relation :

$\{ (d_1, d_2),(d_2, d_3) \}$.

Consider now the formula : $(P(x, y) ∧ P(y, z)) → P(x, z)$ [call it : (B)].

If we "assign" to $x$ the "value" $d_1$ and to $y$ the "value" $d_2$ and finally to $z$ the "value" $d_3$, we have that:

$P(d_1,d_2) \land P(d_2, d_3)$ is true,

while :

$P(d_1, d_3)$ is false.

Thus, we have a conditional: $\text{true } \to \text { false}$ that is $\text {false}$, for the said values.

Having found in the interpretation some values for $x,y,z$ that "falsify" the formula (B), we have shown that it is not true for all possible values, and thus the formula (A) is false in that interpretation.