Question

Mathematical Proofs

Answer 1

nice

Answer 2

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

Answer 3

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

Answer 4

What is this

Answer 5

Proofs.

Answer 6

Teach me

Answer 7

What does <−−> mean?

Answer 8

its the symbol for if and only if.

Answer 9

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?

Answer 10

Yeah I'd also like to know.

Answer 11

What it says after R?

Answer 12

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.

Answer 13

Oh, I had no idea!

Answer 14

Thanks for the tip :)

Answer 15

Yes after R

Answer 16

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

\[p \Rightarrow q \quad \Longleftrightarrow \neg \;p \;\;\text{V}\;\;q\]

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

LOL i just realized that I mistyped that. Sorry!

Answer 23

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 \(x \neq z\).

Answer 24

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 \(0/x=0\).

Answer 25

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.

Answer 26

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

Answer 27

Practice, and a little bit of natural skill is how I'm good.

I've also had some amazing teachers.

Answer 28

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

Answer 29

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

Answer 30

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

Answer 31

Sounds good. :)