Mathematical Proofs

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Mathematical Proofs

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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nice
Prove that \[\exists z \in \mathbb{R} \forall x \in \mathbb{R}^+[\exists y \in \mathbb{R}(y-x+y/x) <--> x \neq z)\]
Lol, sorry for the wait, it was a pain to type up.

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What is this
Proofs.
Teach me
What does <−−> mean?
its the symbol for if and only if.
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?
Yeah I'd also like to know.
What it says after R?
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.
Oh, I had no idea!
Thanks for the tip :)
Yes after R
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 \[p \Rightarrow q \quad \Longleftrightarrow \neg \;p \;\;\text{V}\;\;q\]
So to show implication to the right, let's see if we can prove the simpler statement.
I'm a little confused about the statement \((y-x+y/x)\). What is it saying? In this form it's virtually meaningless.
That's a good question.. so there exists y in R(y - x = y/x) iff x does not equal z.
This is just one confusing statement that should not be legal to give to student. Just saying.
There's supposed to be an equals sign there. That helps. Give me a second to think about this.
LOL i just realized that I mistyped that. Sorry!
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 \(x \neq z\).
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 \(0/x=0\).
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.
KingGeorge. You are amazing. You are seriously my hero. I don't know how you are so good at this, but thank you.
Practice, and a little bit of natural skill is how I'm good. I've also had some amazing teachers.
I know who I'm asking for help on proofs from now on :). I just don't know how I can reward you..
As long as you're trying to learn, I'll be good.
Well, if you're ever in Seattle, I'll buy you dinner.
Sounds good. :)

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