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$\exists x \exists y (x\geq y)$

The universe of discourse is all real numbers. This says that there exists an $x$ and there exists a $y$ such that $x\geq y$. But what is this actually trying to say? I would like to think that it's saying "a number is as large as itself", but I would like some clarification.

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The statement evaluates to true. I don't think it has much more merit than that. –  Tunococ Jan 28 '13 at 13:43
    
If you want to say that every number is as large as itself, that would be $\forall x (x\geq x)$. –  Michael Greinecker Jan 28 '13 at 13:55

2 Answers 2

up vote 2 down vote accepted

But what is this actually trying to say?

This ($\exists x \exists y (x\geq y $)) is actually just saying that there is a real number $x$ such that there is a real number $y$ such that $x$ is greater than or equal to $y$.

Note that technically the $y$ could (will) depend on the $x$. So the specific choice of $y$ might depend on $x$. This makes sense because if $x =3$, then you could pick $y = 2$, but if $x =1$, then you can't pick $2$, but you obviously need a number less than or equal to $1$, say $y = 0$.

Now one might also think of the statement as: there are two real numbers such that one is greater than or equal to the other. Or: There are two real numbers such that one is not greater than the other.

If you want to say that

a number is as large as itself

Then you are saying that given an $x$, this number is "as large as it self". The "as large as it self" might be a bit confusing to some. But one might interpret this to mean that $x\geq x$. So one could write the statement like: $\forall x (x\geq x)$. I would disagree with the saying that the statement is $\exists x (x\geq x)$ because this statement is just saying that there is just some $x$ such that $x$ is "as large as it self". It doesn't say that any real numbers are "as large as it self". The "a number" part I would interpret as "for all numbers". Also note that in this statement (a number is as large as itself) we are only talking about one number.

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I think $y$ does not depend on $x$ in $\exists x\exists y$ which is equivalent to $\exists y\exists x$ but depends in $\forall x\exists y$. –  user59671 Jan 30 '13 at 2:44
    
@CutieKrait: Note that when you write $\exists x \exists y$, then $y$ could (as I write) depend on $x$. in face in this case $y$ will depend on $y$. Given $x$ you can find $y$ such that $x\geq y$. And if you start with a different $x$, then you might need a different $y$. –  Thomas Jan 30 '13 at 2:52

$(\exists x)(\exists y)(x\geq y)$ means there are two real numbers (possibly the same) such that one is not smaller than the other.

$(\exists x)(x\geq x)$ means there is a real number not smaller than itself.

$(\forall x)(x\geq x)$ means every real number is not smaller than itself.

Each of the three statements above happens to be true.

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