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I having a lot of trouble converting two english statements to predicate logic:

  1. For every positive real number x, there is some real number y that is the square root of x (extend this statement to show that there are two distinct square roots of x)

  2. For any two distinct real numbers there is a third real number with a value between the first two.

I figured out the first part of the first statement : ∀x (x > 0 → ∃y (y=$\sqrt{x}$)) but I am unable to solve the second part (extend this statement to show that there are two distinct square roots of x)

and I am clueless how to go about the second question (For any two distinct real numbers...)

Any help will be greatly appreciated!!

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I give you a hint:

You can reformulate the parenthesized sentence into: For any positive real $x$ there exist numbers $y_1$ and $y_2$ such that they are not equal and such that each of them is a square root of $x$. I would write "$y$ being a square root of $x$" as $y^2=x$, since $\sqrt{x}$ is not unique in your statement.

For the second part: For any two $x_1$ and $x_2$ which are not equal there exists $y$ such that ($x_1$ is smaller than $y$ and $y$ is smaller than $x_2$) or ($x_2$ is smaller than $y$ and $y$ is smaller than $x_1$).

Is it clear to you why these are the same?

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  • $\begingroup$ I think so... thank you $\endgroup$ Jun 9 '21 at 16:16

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