Order of logical quantifiers

Question

I don't understand the difference between these $2$ statements:

$1)$ $(\forall x\in \Bbb R)(\exists y\in\Bbb R)x+17=y$

$2)$ $(\exists x\in\Bbb R)(\forall y\in\Bbb R) x+17=y$

The first one should be true, and the second one should be false. I understand that the first one should be read as "For all real numbers $x$, there is a real number $y$ such that $x+17=y$. I understand that no matter which $x$ I pick, there will always be $y$ that satisfies that equation. However, I don't understand why the second one is false. I would read it as: "There is a real number $x$ for all real numbers $y$ such that $x+17=y$". In my view, that has the same meaning as the first statement, since no matter which $y$ I pick there will exist $x$ that satisfies that equation.

Answer 1

The second sentence should be read as "There is a real number $x$ so that for every real number $y$ $x + 17 = y$". The "so that" is critical - we're saying that "for every real number $y$, $x + 17 = y$" is a property that $x$ has. By contrast, in your reading of the sentence, "there is a real number $x$ for every real number $y$" makes $x$ belong to $y$, not the other way around; this lets you choose a different $x$ for each $y$. In the correct reading of the sentence, you must choose a single $x$ so that this one $x$ works for every single $y$.

Answer 2

1) "For all real numbers $x$, there exists a real number $y$ such that $x+17=y$"

This is obviously true.

2) "There exists a real number $x$ such that for all real numbers $y$, $x+17=y$"

This is false. If we have two different numbers $y_1$ and $y_2$, we can't have both: $$x+17=y_1$$ $$x+17=y_2$$ as this would imply $y_1=y_2$, which contradicts our definition of $y_1$ and $y_2$.

Answer 3

$1)$ $(\forall x\in \Bbb R)(\exists y\in\Bbb R)x+17=y$

For every real x, there is some real y that is 17 more than x. It may be a different y for each x.

$2)$ $(\exists x\in\Bbb R)(\forall y\in\Bbb R) x+17=y$

There is some real x that is 17 less than every real y. It is the same x for every y.

Answer 4

When you say $\exists x$, you fix the variable say $x=a$. Once you have fixed it, now you have $(\exists a)( \forall y\in\mathbb{R}) a+17=y.$ This should be true for all $y.$ If $y=18$ then $a=1$ but if $y=19$ then $a=2$. But $a$ is fixed. Hence the second statement is false.