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The statement:

$\forall a,b \in \mathbb{R}, \exists c,d \in \mathbb{R}$, such that if $ab\geq cd$, then $a\geq c$ and $b\geq d$.

First, does this mean that $a, b, c, d$ all have to be different real numbers or can they all be the same?

Also, Does this mean that for all combinations of $a, b$ we can find a $c, d$ such that etc? Or does it mean that for all combinations of $a, b$ there are 2 set $c, d$ real numbers such that etc?

I don't know if I can just take $a=c$ and $b=d$ to prove it or not.

Alain Remillard
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etkachuk
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1 Answers1

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There is no requirement that the numbers be distinct. For example, the sentence $$\forall x\exists a\exists b\exists c(x=a=b=c)$$ is true (given $x$, let $a=b=c=x$). This is similar to how "or" in logic is understood inclusively.

In particular, the sentence in question is true, as you say, by taking $a=c$ and $b=d$. (And that leads me, incidentally, to suspect there might be a typo ...)

Noah Schweber
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