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kevo

Mathematical Proofs

  • one year ago
  • one year ago

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  1. mr.luna
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    nice

    • one year ago
  2. kevo
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    Prove that \[\exists z \in \mathbb{R} \forall x \in \mathbb{R}^+[\exists y \in \mathbb{R}(y-x+y/x) <--> x \neq z)\]

    • one year ago
  3. kevo
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    Lol, sorry for the wait, it was a pain to type up.

    • one year ago
  4. Wheaton71
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    What is this

    • one year ago
  5. kevo
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    Proofs.

    • one year ago
  6. Wheaton71
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    Teach me

    • one year ago
  7. eliassaab
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    What does <−−> mean?

    • one year ago
  8. kevo
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    its the symbol for if and only if.

    • one year ago
  9. eliassaab
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    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?

    • one year ago
  10. Wheaton71
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    Yeah I'd also like to know.

    • one year ago
  11. kevo
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    What it says after R?

    • one year ago
  12. KingGeorge
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    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.

    • one year ago
  13. kevo
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    Oh, I had no idea!

    • one year ago
  14. kevo
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    Thanks for the tip :)

    • one year ago
  15. eliassaab
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    Yes after R

    • one year ago
  16. KingGeorge
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    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 \[p \Rightarrow q \quad \Longleftrightarrow \neg \;p \;\;\text{V}\;\;q\]

    • one year ago
  17. KingGeorge
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    So to show implication to the right, let's see if we can prove the simpler statement.

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

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

    • one year ago
  20. kevo
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    This is just one confusing statement that should not be legal to give to student. Just saying.

    • one year ago
  21. KingGeorge
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    There's supposed to be an equals sign there. That helps. Give me a second to think about this.

    • one year ago
  22. kevo
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    LOL i just realized that I mistyped that. Sorry!

    • one year ago
  23. KingGeorge
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    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 \(x \neq z\).

    • one year ago
  24. KingGeorge
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    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 \(0/x=0\).

    • one year ago
  25. KingGeorge
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    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.

    • one year ago
  26. kevo
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    KingGeorge. You are amazing. You are seriously my hero. I don't know how you are so good at this, but thank you.

    • one year ago
  27. KingGeorge
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    Practice, and a little bit of natural skill is how I'm good. I've also had some amazing teachers.

    • one year ago
  28. kevo
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    I know who I'm asking for help on proofs from now on :). I just don't know how I can reward you..

    • one year ago
  29. KingGeorge
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    As long as you're trying to learn, I'll be good.

    • one year ago
  30. kevo
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    Well, if you're ever in Seattle, I'll buy you dinner.

    • one year ago
  31. KingGeorge
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    Sounds good. :)

    • one year ago
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