anonymous
  • anonymous
Mathematical Proofs
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
nice
anonymous
  • 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
  • anonymous
Lol, sorry for the wait, it was a pain to type up.

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anonymous
  • anonymous
What is this
anonymous
  • anonymous
Proofs.
anonymous
  • anonymous
Teach me
anonymous
  • anonymous
What does <−−> mean?
anonymous
  • anonymous
its the symbol for if and only if.
anonymous
  • 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
  • anonymous
Yeah I'd also like to know.
anonymous
  • anonymous
What it says after R?
KingGeorge
  • 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
  • anonymous
Oh, I had no idea!
anonymous
  • anonymous
Thanks for the tip :)
anonymous
  • anonymous
Yes after R
KingGeorge
  • 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
  • KingGeorge
So to show implication to the right, let's see if we can prove the simpler statement.
KingGeorge
  • 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
  • anonymous
That's a good question.. so there exists y in R(y - x = y/x) iff x does not equal z.
anonymous
  • anonymous
This is just one confusing statement that should not be legal to give to student. Just saying.
KingGeorge
  • KingGeorge
There's supposed to be an equals sign there. That helps. Give me a second to think about this.
anonymous
  • anonymous
LOL i just realized that I mistyped that. Sorry!
KingGeorge
  • 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
  • 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
  • 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
  • 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
  • KingGeorge
Practice, and a little bit of natural skill is how I'm good. I've also had some amazing teachers.
anonymous
  • 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
  • KingGeorge
As long as you're trying to learn, I'll be good.
anonymous
  • anonymous
Well, if you're ever in Seattle, I'll buy you dinner.
KingGeorge
  • KingGeorge
Sounds good. :)

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