Determine whether $∀x∈ℝ,∃y∈ℝ$ such that $x+y=0$ & $∃x∈ℝ,∀y∈ℝ$ such that $x+y=0$ is true or false.

Question

Please help me out of this I am easily confuse by this kind of question.

Determine whether $∀x∈ℝ,∃y∈ℝ$ such that $x+y=0$ is true.

Logical thinking I know that it is true because for all $x$, I can choose a corresponding $y$ that will satisfy $x+y=0$. But should it be $x = -y$ or $ y =-x$?

Determine whether $∃x∈ℝ,∀y∈ℝ$ such that $x+y=0$ is true. I will say that it is false because there only one $y$ that can make $x+y=0.$

Thanks!

Answer 1

For the first part, the equations $x=-y$ and $y=-x$ are actually equivalent, but what you need to do is:

Given a certain value of $x$, you need to find the appropriate value of $y$ such that $x+y=0$.

So, your result must be something that defines what value of $y$ you are taking. For example, if $x=4$, what value of $y$ do you take?

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Second part: Your intuition is correct, now you need to put that into a rigorous proof. That is, you are disproving the statement:

$\exists x\in\mathbb R:\forall y\in\mathbb R: x+y=0$

Which means you must prove the negation of this statement, which is:

$\forall x\in\mathbb R: \exists y\in\mathbb R: x+y\neq 0$

To prove this statement, you must pick an arbitrary $x$ and find some $y$ for which $x+y$ is not equal to $0$.