1. ## Translating Quantified statements.

Hello,

I am asked to translate both $\forall x \exists y Q(x,y)$ and $\exists y \forall x Q(x,y)$, where $Q(x,y)$ is a propositional function that represents the statement, "x has sent an e-mail message to y," where the domain for both x and y consists of students in one's class.

I feel as though the two quantified statements would translate into the same English statement, aren't both statements the same?

2. ## Re: Translating Quantified statements.

Originally Posted by Bashyboy
I am asked to translate both $\forall x \exists y Q(x,y)$ and $\exists y \forall x Q(x,y)$, where $Q(x,y)$ is a propositional function that represents the statement, "x has sent an e-mail message to y," where the domain for both x and y consists of students in one's class.
$\forall x \exists y Q(x,y)$ translates: "Every student in class sends email to some student in class."

$\exists y \forall x Q(x,y)$ translates: "Some student in class is sent emails by everyone in class."
Note the change in 'voice'.

3. ## Re: Translating Quantified statements.

Okay, so they have the same meaning, but the statements are translated in the order of the quantifiers

4. ## Re: Translating Quantified statements.

Originally Posted by Bashyboy
Okay, so they have the same meaning, but the statements are translated in the order of the quantifiers
Are you kidding? They do not have the same meaning at all.

The first says that everyone emails. The second says someone receives an email everyone.

5. ## Re: Translating Quantified statements.

This become more obvious if Q(x, y) means that y is the mother of x. Then ∀x ∃y Q(x, y) means that every person has (his or her own) mother. In contrast, ∃y ∀x Q(x, y) means that there exists a single mother of everyone.