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I have to write out what the following statements mean and whether they are true or false:

  1. $(∃y ∈ N)(∀x ∈ N)(x < y)$

  2. $(∀y ∈ N)(∃x ∈ N)(x < y)$

  3. $(∀x ∈ N)(∀y ∈ N)(x < y)$

So for number 1, would I say that there exists $y \in N$ that is greater than all values of $x \in N$ ? Would this be going in the right direction? How would I know if this is true or not?

Michael Turner
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    You're right about the first one, but, in english, it's a bit more natural to say: there exists a natural number greater than all natural numbers. With this rephrasing, its trutfulness (or lack thereof) should become apparent. – Git Gud Oct 20 '13 at 15:40
  • Ok thanks. So for 2) There exists a natural number less than all natural numbers. Would this be the wrong way round? What would I do for 3 since both statements are the same? All natural numbers are less than all natural numbers!? – Michael Turner Oct 20 '13 at 15:45
  • Your translation of 3. is correct: all natural numbers are smaller than every natural number. You got 2. wrong. It starts with $\forall$, yet you're starting your translation with 'there exists'. – Git Gud Oct 20 '13 at 15:46
  • Right I thought that would be going about it the wrong way. So all natural numbers are greater than some natural numbers? – Michael Turner Oct 20 '13 at 15:48
  • That's right.${}$ Edit: please consider writing an answer yourself so the question doesn't come up as unanswered. – Git Gud Oct 20 '13 at 15:48
  • Will do. So only the first statement is true? – Michael Turner Oct 20 '13 at 15:52
  • The statement only 1. is true is false. – Git Gud Oct 20 '13 at 15:54
  • So all statements are false? – Michael Turner Oct 20 '13 at 15:55
  • All statements are false. – Git Gud Oct 20 '13 at 15:56

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