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∃ x ∈ R, ∀ y ∈ R x ≥ y

Write the statement in English. A complete answer will not use any mathematical notation, nor the symbols x and y. Write down the truth value of the statement. Write down the negation of the statement in symbols and in English.

my soln:

some real numbers are greater or equal to all real numbers. false noreal numbers are greater or equal to all real numbers

Pls help

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    $\begingroup$ Your version is fine. More idiomatic, in English, would be something like "There is a largest real number." $\endgroup$ – André Nicolas Mar 20 '14 at 2:52
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    $\begingroup$ You are almost correct, although I would say "some real number is greater or equal to all real numbers." Or more precisely, "there is/exists a real number..." $\endgroup$ – 4ae1e1 Mar 20 '14 at 2:52
  • $\begingroup$ The statement translates as: There exists some other real number for all real numbers such that the first is greater than or equal to the latter The statement is false. $\endgroup$ – user122283 Mar 20 '14 at 3:12
  • $\begingroup$ what about the negation? is mine ok? $\endgroup$ – logicunix Mar 20 '14 at 3:38
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    $\begingroup$ Might as well write the formal negation as well and then see if your sentence corresponds to it. The negation is, $\neg(\exists x\in\mathbb R, \forall y\in \mathbb R, x\geq y)$. Recall that $\neg(\exists x,\phi(x))=\forall x,(\neg\phi(x))$ and $\neg(\forall x,\phi(x))=\exists x,(\neg \phi(x)$. Then the above negation becomes $\forall x\in \mathbb R, \exists y\in\mathbb R, x< y$. Is it true that if someone hands you a real number, you can give a real number greater than it? $\endgroup$ – Rachmaninoff Mar 20 '14 at 3:51
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There exists a real number $x$, such that, for every real number $y$, $x$ is either greater than $y$ or equal to $y$. In other words, there is a real number which is greater than or equal to all real numbers.

This statement is false of course.

Proof by contradiction:

  • Assume that there is such number $x$.
  • Observe the number $y=x+1$.
  • Obviously, $x$ is neither greater than $y$ nor equal to $y$.
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$\exists(x \in \mathbb{R})[\forall (y \in \mathbb{R} ) [x \geq y]]$

It means:

There exists a number in reals for which all real numbers are smaller or equal to it.

It's $\infty$, of course.

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    $\begingroup$ $\infty$ is not a real number. Saying "It's $\infty$" is not accurate. $\endgroup$ – N8tron May 25 '14 at 12:32

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