I am to Write the sentence in words that corresponds to the following symbolic statement. Find the negation both symbolically and in words:
∀x ∃y ∋ y > x
For all x: P(x) is true and there is at least one y such that P(y) is true For every element P(y) > P(x)
There exist at least one x such that P(x) is true and For all y: P(y) is true For every element P(y)≤ P(x)
My browser shows this formula as $\forall\exists y\ni y>x$. This is not a well-formed formula because $\ni$ is a binary relation requiring two arguments. That is, $B\ni a$ (more commonly written as $a\in B$ means that $a$ is an element of a set $B$. In fact, I don't believe $\ni$ should be a part of your formula at all. You probably meant $\forall x\exists y\;y>x$.
This does not make much sense because $P$ was not defined. Every object you mention must be properly introduced.
I agree.
This statement does not properly introduce $x$. It is not clear whether this holds for all $x$ or for some $x$. In addition, this sentence switches $x$ and $y$ compared to the original statement, which introduces an additional difficulty. Usually one constructs the negation of $\forall x\exists y\;P(x,y)$ as follows: $\exists x\forall y\;\neg P(x,y)$, i.e., the quantifiers change, but the variable names don't. Of course, the latter formula is equivalent to $\exists y\forall x\;\neg P(y,x)$, but again, this renaming introduces extra complexity. It seems that you got confused by this extra complexity because you renamed $x$ and $y$ that follow the quantifiers, but did not rename them in the rest of the formula. Finally, the negation of a strict inequality $y>x$ is a non-strict inequality.
Yes, this is incorrect. The correct negation is
There exists an y such that for all x, x is less than or equal to y,
or
There exists an x such that for all y, y is less than or equal to x.
That is, there exists a greatest element.
The original formula does not have a capital $Y$. Again, $x$ is not properly introduced by a quantifier here. It looks like $\exists$ pertains also to $x$, but this is not correct.