# Determining and justifying the validity of an argument

Context: Question made up by uni lecturer

Original statement: There exists two positive real numbers $$x$$ and $$y$$ such that for all positive integers $$z$$, $$\frac{x}{y}>z$$.

So the question was to find the negation of the statement, and then determine whether the original statement or its negation was true.

I found its negation to be: For all positive real numbers $$x$$ and $$y$$, there exists a positive integer $$z$$ such that $$\frac{x}{y}\le z$$.

The lecturer's solution to the question says that the negation is true since for any positive reals $$x$$ and $$y$$, you can choose $$z$$ to equal the ceiling of $$\frac{x}{y}$$.

When I attempted the question myself, I said that the original statement is true because you can take $$x=z+1$$ (which would be a positive integer that still belongs to the set of all positive real numbers) and $$y=1$$ (which is a positive real number), as $$\frac{x}{y}=\frac{z+1}{1}=z+1>z$$.

Thanks

The problem with your reasoning is the order. There exist $$x,y$$ positive integers such that for all positive integers $$z$$ we have $$\frac{x}{y}>z$$, so first you must pick an $$x$$ and a $$y$$, and then you must test whether $$\frac{x}{y}>z$$ for every positive integer $$z$$. Therefore you can't define $$x=z+1$$ as when you pick $$x$$ you don't know $$z$$ yet.

"There exists two positive real numbers $$x$$ and $$y$$ such that for all positive integer $$z$$ : $$\dfrac x y > z$$"

The intuition says that it is False. $$\dfrac x y$$ is a positive real; thus, the statement amounts to asserting that there is a real that is greater than every integer, which is not.

You reasoning is wrong because you have swapped the choice of the numbers : you start from $$z$$ and choose $$x$$ and $$y$$ accordingly.

The negation of the original statement is : $$\forall x \ \forall y \ \exists z \ (\dfrac y y \le z)$$.

Thus, choose $$x$$ and $$y$$ positive whatever and what you get is a new positive real $$\dfrac x y$$.

Now, you have to choose an integer $$z$$ (obviously positive) that is greater-or-equal to $$\dfrac x y$$.

And this must be always possible, because $$\dfrac x y$$ is a number $$r.r_1 r_2 r_3 \ldots$$.

Consider as $$z$$ the number $$r+1$$ and it's done.