## kevo Group Title Mathematical Proofs 2 years ago 2 years ago

1. mr.luna Group Title

nice

2. kevo Group Title

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

3. kevo Group Title

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

4. Wheaton71 Group Title

What is this

5. kevo Group Title

Proofs.

6. Wheaton71 Group Title

Teach me

7. eliassaab Group Title

What does <−−> mean?

8. kevo Group Title

its the symbol for if and only if.

9. eliassaab Group Title

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?

10. Wheaton71 Group Title

Yeah I'd also like to know.

11. kevo Group Title

What it says after R?

12. KingGeorge Group Title

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.

13. kevo Group Title

14. kevo Group Title

Thanks for the tip :)

15. eliassaab Group Title

Yes after R

16. KingGeorge Group Title

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

17. KingGeorge Group Title

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

18. KingGeorge Group Title

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

19. kevo Group Title

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

20. kevo Group Title

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

21. KingGeorge Group Title

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

22. kevo Group Title

LOL i just realized that I mistyped that. Sorry!

23. KingGeorge Group Title

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 $$x \neq z$$.

24. KingGeorge Group Title

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 $$0/x=0$$.

25. KingGeorge Group Title

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.

26. kevo Group Title

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

27. KingGeorge Group Title

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

28. kevo Group Title

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

29. KingGeorge Group Title

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

30. kevo Group Title

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

31. KingGeorge Group Title

Sounds good. :)