Understanding Scope of Quantifiers

Question

I have a set of questions below that I'm trying to logically work my way through. I have ideas about each but I'm not sure if I'm correct. Any advice would be most welcome! Thanks so much!

1) I know that the converse is not true because there exists a Y not all Y. I don't know how to write out the converse though. 2) I believe they differ in meaning because the ∀x [P(x)]∨ ∀x [Q(x)] could have two different possible values for x. I'm not sure of a good example of that though. 3) would be similar to #2 where the first half could have two different values for X. Also not sure of an example. 4) I'm not sure at all about this one. I believe they are equivalent because the domain is not changing.

1. (6) Explain why [∀y∀x P(x, y)] → [∀x∃y P(x, y)]. Use a counterexample to demonstrate that the converse is not always true.
2. (4) How do ∀x [P(x)∨Q(x)] and ∀x [P(x)]∨ ∀x [Q(x)] differ in their meaning? Use examples to illustrate.
3. (4) How do [∀x P(x)] → [∀x Q(x)] and ∀x [P(x) → Q(x)] differ in their meaning? Use examples to illustrate.
4. (4) Are ∀x [P(x) → ∃y [Q(x, y)]] and ∀x∃y [P(x) → Q(x, y)] logically equivalent? Explain why or why not.

Answer 1

In finding examples, it is often useful to restrict $P$ and $Q$ to some set. For example: $P(x) \equiv (x \in \Bbb N \implies 2\mid x)$.

1. You are just looking for some statement $P(x, y)$ which for any $x$ has a value $y$ which makes it true, but which is not true for every $y$. For example, $x = y$.
2. $\forall x [P(x) \vee Q(x)]$ means for each value of $x$, either one or the other of $P$ or $Q$ is true. $\forall x [P(x)] \vee \forall x [Q(x)]$ means that either $P$ is true for every $x$, or $Q$ is true for every x. Look for an example where $P$ and $Q$ complement each other: one is true in the places where the other is false.
3. is indeed similar to 2. The first form is true any time that $P$ is false for even one $x$, regardless of $Q$. The second form requirs $Q$ to be true for each $x$ where $P$ is false.
4. The change between $\forall x [P(x) \implies \exists y [Q(x,y)]]$ and $\forall x \exists y [P(x) \implies Q(x,y)]$ is the scope of the $\exists y$, which in the first only includes "$Q(x,y)$", but in the second also includes "$P(y) \implies$". Now ask yourself, what effect does knowing a particular value of $y$ have on the meaning of "$P(x)\implies$"?