Mathematical Proofs

- anonymous

Mathematical Proofs

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- schrodinger

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- anonymous

nice

- anonymous

Prove that \[\exists z \in \mathbb{R} \forall x \in \mathbb{R}^+[\exists y \in \mathbb{R}(y-x+y/x) <--> x \neq z)\]

- anonymous

Lol, sorry for the wait, it was a pain to type up.

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## More answers

- anonymous

What is this

- anonymous

Proofs.

- anonymous

Teach me

- anonymous

What does <−−> mean?

- anonymous

its the symbol for if and only if.

- anonymous

It is still not clear to me what the last statement mean.
\[
\exists z \in \mathbb{R} \forall x \in \mathbb{R}^+[\exists y \in \mathbb{R}
\]
such that what?

- anonymous

Yeah I'd also like to know.

- anonymous

What it says after R?

- KingGeorge

just fyi, if you want to increase the space between characters in the equation editor, simply type "\;" for a small space, "\:" for a slightly bigger one, "\quad" for a big one, and "\qquad" for a giant one.
This helps increase readability.

- anonymous

Oh, I had no idea!

- anonymous

Thanks for the tip :)

- anonymous

Yes after R

- KingGeorge

I think the best way to approach this would be to split it into two parts. First we want to show implication tot he right, and second we want to show implication to the right. Also note that
\[p \Rightarrow q \quad \Longleftrightarrow \neg \;p \;\;\text{V}\;\;q\]

- KingGeorge

So to show implication to the right, let's see if we can prove the simpler statement.

- KingGeorge

I'm a little confused about the statement \((y-x+y/x)\). What is it saying? In this form it's virtually meaningless.

- anonymous

That's a good question.. so there exists y in R(y - x = y/x) iff x does not equal z.

- anonymous

This is just one confusing statement that should not be legal to give to student. Just saying.

- KingGeorge

There's supposed to be an equals sign there. That helps. Give me a second to think about this.

- anonymous

LOL i just realized that I mistyped that. Sorry!

- KingGeorge

Let's show implication to the right first. To show this, we need to choose a z such that for all y and x (x positive) \((y-x =y/x)\) or \(x\neq z\).
Just choose z to be negative. Since x is positive, we know that \(x \neq z\).

- KingGeorge

Now we need to show implication to the left. To show this, we need to choose a z such that for all positive x, there exists a y such that that \(z=x\) or that \(y-x\neq y/x\)
Here, just choose \(y=0\). Since x is positive, \(0-x\) is less than 0, and \(0/x=0\).

- KingGeorge

Therefore, we are done. Sorry that took a while for me to write.
Also, Instead of writing "we need to choose a z such that for all y and x (x positive)" in the first part, I should have written "we need to choose a z such that for all positive x there exists a y"
It doesn't really matter in the end however.

- anonymous

KingGeorge. You are amazing. You are seriously my hero. I don't know how you are so good at this, but thank you.

- KingGeorge

Practice, and a little bit of natural skill is how I'm good.
I've also had some amazing teachers.

- anonymous

I know who I'm asking for help on proofs from now on :). I just don't know how I can reward you..

- KingGeorge

As long as you're trying to learn, I'll be good.

- anonymous

Well, if you're ever in Seattle, I'll buy you dinner.

- KingGeorge

Sounds good. :)

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