Logical Connectiveness

Question

$∃b∈S2 ∀a∈S1 :p(a,b)$

This is my attempt: $(x,y)∈ $ {$[ (a,c)∧ (b,c) ] ∨ [(a,d)∧ (a,d) ] $} | (P(x,y)

Answer 1

$$\forall x\in M_1:P(x,y)$$ is $$P(a,y)\land P(b,y),$$

and

$$\exists y\in M_2:\forall x\in M_1:P(x,y)$$ is $$(P(a,c)\land P(b,c))\lor(P(a,d)\land P(b,d)).$$

Answer 2

Not exactly...

In finite domains, $\exists$ acts as a disjunction while $\forall$ acts as a conjunction.

Thus, set $y=c$; we have that $(\forall x \in M_1) P(x,y)$ will be:

$P(a,c) \land P(b,c)$.

Now, do the same for $y=d$.