Originally Posted by

**matthayzon89** I need to Prove or Disprove: There does not exist a real number x such that for all real numbers y, xy=1.

This is false. I am going to try to prove it using contradiction.

You mean: "this is true", i.e.: the claim is true, since there indeed is NOT such a number x

Proof:

Suppose not. That is, suppose that there **does** exist a real number x such that for all real numbers y, xy=1.

You mean "suppose yes: there is such a number..." . I think you got confused with so many negations.

Let x be any real number r. X=1/r.

So you define $\displaystyle x=r\,,\,X=\frac{1}{r}=\frac{1}{x}$ ... why to choose the same letter, capital and non-capital, twice? This may help to make things even more confusing.

Let y be an real number r. Y=r/1

And now you've defined so far $\displaystyle x=y=r\,,\,\,X=Y=\frac{1}{r}$ ...is this really what you meant?

Then,

X*Y= (1/r)*(r/1) by substitution

= 1 By algebra.

This is wrong: $\displaystyle XY=\frac{1}{r}\frac{1}{r}=\frac{1}{r^2}$ ...and that's pretty much there is to it.

This is a contradiction because there **does** exist a real number x such that for all real numbers y x*y=1. **end of proof**

There is no proof at all above of anything. Perhaps you could try the following:

Suppose there exists such a real number $\displaystyle x\,\,\,s.t.\,\,\,xy=1\,\,\forall\,y\in\mathbb{R}$ . Choose now $\displaystyle y=2\Longrightarrow$[tex]

$\displaystyle 2x=1\Longrightarrow x=\frac{1}{2}$ , and now choose say $\displaystyle y=3$ and get that x must equal two different numbers: contradiction and we're done.

Tonio

Can some someone please let me know if this is a correct way to write it on paper? I am pretty confident, however I would like to make sure, is there anything I can add for correctness?

Thank You,

Matt H.