I want to know how to properly write below statements using quantifiers and simple relations ($\in, =, <, \leq, |$):
- Between any 2 rational numbers exists number different than these 2.
- Every natural number greater than 1 has at least one prime factor.
1st statement seems to be easy, I think it's right: $\forall x, y\in \mathbb{R} \exists c\in \mathbb{R}: x < c < y \lor y < c < x$
but I have problems writing properly 2nd statement. I tried something like that, but I think it's wrong:
$\forall n\in\mathbb{N}, n > 1 \exists p \in \mathbb{N} : p|n \land \forall x \in \mathbb{N} x < p : x\not| p \land p \not= 1$
Also, I'd be interested how to write "definition" of NWD(a,b) using quantifiers and basics relations. Can you point me to some books/tutorials where I can excercise "rewriting" statemants in natural language to mathematics language?
Thanks for help