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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

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  • $\begingroup$ by the way x and y represent real numbers $\endgroup$ – Mac Oct 9 '13 at 19:06
  • $\begingroup$ 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. $\endgroup$ – Jürgen A. Erhard Aug 11 '16 at 17:09
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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).

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  • $\begingroup$ Thanks for the help. I'll make sure to make appropriate changes. However when you say in c) Since there is no largest real number, the sentence is call...what does that mean? Also should I assume my true/false placements are correct as you did not really comment about those? $\endgroup$ – Mac Oct 9 '13 at 19:20
  • $\begingroup$ @Mac: "call" was a really weird typo for "false". The rest of your answers look fine at first glance. $\endgroup$ – Henning Makholm Oct 9 '13 at 19:22

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