- #1

oferon

- 30

- 0

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:

[itex]\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[/itex]

Only then I checked the solution given by my teacher said:

[itex]\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[/itex]

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:

[itex]\exists x,y\in R.[(x<y)\wedge (x^2>y^2)][/itex]

Thanks in advance for your time :)