The difference between $\forall x \in \Bbb{R}, \exists y \in \Bbb{R},\, x + y > 0$ and $\exists x \in \Bbb{R}, \forall y ∈ \Bbb{R}, x + y > 0$ I'm sure its a very basic question but I can't understand why the first of these statements is true and the second is false as the professor says.
1: $$\forall x \in \mathbb{R}, \exists y \in \mathbb{R},\, x + y > 0$$
2: $$\exists x \in \mathbb{R}, \forall y ∈ \mathbb{R}, x + y > 0$$
I know the first is true as you write:
$$y = 1 − x \in \mathbb{R}$$
I don't however understand why I cant do the reverse for the second and write:
$$x = 1 − y \in \mathbb{R}$$
He says that the second is false because its negation is true, the negation being:
$$\forall x \in \mathbb{R}, \exists y \in \mathbb{R}, x + y ≤ 0$$
I understand that this is true as you can write:
$$y = −x \in \mathbb{R}$$
I also understand that is a negation is true then the statement should be false but I cant understand how the two original statements differ?
 A: The second statement says that there is some single real number $x$ with the following property: no matter what real number you add to $x$, the sum is positive. This is clearly false, because whatever $x$ is, $-x$ is a real number, and adding $-x$ to $x$ does not yield a positive sum.
The first is a much weaker statement. It does not say that there is a single real number $x$ whose sum with every real number is positive. It says that for each real number $x$ there is at least one real number $y$ whose sum with $x$ is positive, and this $y$ can be different for different values of $x$.
To put it a bit differently, the first statement says that no matter what $x\in\Bbb R$ you choose, I can find a $y\in\Bbb R$ such that $x+y>0$, but my choice of $y$ can depend on what $x$ you give me. The second says that I can find an $x\in\Bbb R$ such that no matter what $y\in\Bbb R$ you choose, the sum $x+y$ will be positive. I have to choose one $x$ first that will ‘work’ no matter what $y$ you then decide to give me, and that’s simply not possible.
A: You don't need the negative for the second statement. Just note that 2) means that exists some $ x $ that works with every $y$. This is false for $ y =-x $ because in this case we have $ x+(-x)=0$.
Also these examples show that you can't "commute" $\forall $ and $\exists $.
A: In plain English, the first statement says, "If you have a real number $x$, then you can always find a real number $y$ such that their sum $x+y$ is positive." This is obviously true: you just take $y$ to be any number bigger than $-x$.
On the other hand, the second statement says, "There is a real number $x$ such that adding any real number $y$ to it will make the sum $x+y$ positive." Is this true? Well the negation is: "If you have a real number $x$, then you can always find a real number $y$ such that their sum $x+y$ is negative or zero." This is true: just take $y$ to be any number less than or equal to $-x$. Since the negation is true, the original statement is false.
A: In the first statement you say that for all elements in $\mathbb{R}$, you can find another element such that their sum is positive.
In the second statement you say that there is a specific element in $\mathbb{R}$ such that the sum with any other element of $\mathbb{R}$ is positive.
The second statement doesn't make any sense, because you don't get to choose the $y$, as you're trying to do. In the first statement, given any $x$, you can find $y$.
