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The statement i want to translate is this: x is the smallest real number and P(x) is false

$\exists x \in \mathbb{R} \forall y \in \mathbb{R} \neg (P(x)); x > y$

I don't know how to put two statements in one predicate sentence.

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    +1 for showing your work and getting the "hang" of formatting!2012-10-30

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You don’t want the existential quantifier: ‘$x$ is the smallest real number’ is simply $\forall y\in\Bbb R\Big(\lnot(y Since ‘$P(x)$ is false’ is $\lnot P(x)$, the conjunction of the two is simply

$\forall y\in\Bbb R\Big(\lnot(y

This says that some $x$ that was presumably specified previously has the desired properties.

The existential quantifier is needed if you want to say that such an $x$ exists:

$\exists x\left(\forall y\in\Bbb R\Big(\lnot(y

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    @Peter: At the relevant level of mathematical sophistication, the English sentence can’t be translated without *some* non-logical background assumptions. You want to make fewer than I did; that’s fine $-$ another day I might feel the same way. Moreover, at this level there are few canonical translations: the acceptability of$a$translation is entirely up to the instructor. I’d be more inclined to take your view of the matter if I thought that the point of this particular exercise was to translate *is the unique$x$such that*, but I don’t: I think that it’s early days still.2012-10-31