Discrete math - simple formalism question

[oferon]

I never used discrete math terms in english before, so I hope it sounds clear enough:

Formalize the following:
1) Between every two different real numbers there is a rational number
2) There exist real numbers x and y, such that x is smaller than y, yet x^2 is bigger than y^2

Now the solution I wrote for 1 is:

$\forall x,y \in R.x\neq y \hspace{5 mm} \exists z\in Q.\left\langle[(x<z)\wedge (y>z)]\vee [(y<z)\wedge (x>z)]\right\rangle$Only then I checked the solution given by my teacher said:

$\forall x,y \in R.x\neq y \Rightarrow \exists z\in Q.\left\langle[(x<z)\wedge (y>z)]\vee [(y<z)\wedge (x>z)]\right\rangle$

Now, where did this implication arrow come from, and is it necessary? If so, what's wrong with my solution then?For the second sentence we didn't get any solution, so I just want to confirm the following is correct:

$\exists x,y\in R.[(x<y)\wedge (x^2>y^2)]$Thanks in advance for your time :)

 

[gustav1139]

The 'implies' is there to emphasize the logical connection there. If you have two reals like that, then you can find such a z. Moreover, if you can't find such a z, then x=y (or one of x,y is not a real number, which seems less likely).
I probably would have left out the arrow as well, since the question is phrased as a statement, and not an implication. In any case, the sentence you wrote would usually be acceptable, but since your prof. wants the arrow in, leave it.

Your second answer looks fine to me, unless you want to put a little "s.t." in between there.

 

[oferon]

Hi gustav
Thanks for your reply
Could you just explain what "s.t." means? I'm not very familiar with the english terms.
Thanks a bunch

 

[HallsofIvy]

s.t. = 'such that'. "There exist x and y such that..." means there exist x and y having the following property.