Logical Symbol Statements True/False?

so I'm working through homework questions for proofs class and unsure if I'm correct in my interpretation. I would really appreciate feedback. The questions states: Write the full meaning in English and decide whether True or False

a) ∀x∀y(x ≥ y)

False: for all x and for all y, there is an x which is greater/equal then y.

b) ∃x∃y(x ≥ y)

True: There exists an x and a y, in which an x is greater/equal then y.

c) ∃y∀x(x ≥ y)

True: There exists a y for all of x, where x greater/equal then y

d) ∀x∃y(x ≥ y)

True: For all x, there is a y, in which x greater/equal then y

e) ∀x∃y(x^2 + y^2 = 1)

False: For all x, there is a y in which x^2+y^2 = 1

f) ∃x∀y(x^2 + y^2 = 1)

False: There exists an x for all y in which x^2+y^2 = 1

• by the way x and y represent real numbers – Mac Oct 9 '13 at 19:06
• Yeah, homework long gone, but... if x∈R ʌ y∈R, then c is False, not True. If they are real numbers but from a finite(!) subset of R, then it's True. IFF they're from the same (sub)set. Mathematics and imprecision really don't mix. That would be the most important bit to learn about math. – Jürgen A. Erhard Aug 11 '16 at 17:09

In (a) you say both "for all $x$" and "there is an $x$". You should have only one specifier: for all $x$ and all $y$, $x$ will be greater than or equal to $y$.
In (c) you accidentally mix up the meaning. It should be: There is a $y$ such that every $x$ is greater than or equal to this $y$. The difference from what you have writen is that $\exists y \forall x$ wants an $y$ that does not depend on $x$; the same $y$ must work for all $x$. Since there is no largest real number, the sentence is false.
In (f) you make the same mistake as in (c). It should be: There is an $x$ such that every $y$ makes $x^2+y^2=1$ hold. (This is false, of course).