[Math] Write statements using quantifiers

logicquantifiers

I want to know how to properly write below statements using quantifiers and simple relations ($\in, =, <, \leq, |$):

  1. Between any 2 rational numbers exists number different than these 2.
  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

Best Answer

Between any 2 rational numbers exists number different than these 2.

$\forall x, y\in \mathbb{Q}, \exists c\in \mathbb{Q}, x < c < y \lor y < c < x$

By the way this statement is false since it allows $x=y$.

Every natural number greater than 1 has at least one prime factor

$\forall n \in \mathbb{N}, n > 1, \exists p \in \mathbb{N}, p \mbox{ prime} \land p \mid n$

$p \mbox{ prime} \Leftrightarrow \forall q \in \mathbb{N}, q \not= 1, q \not= p, q \nmid p$

So you get

$\forall n \in \mathbb{N}, n > 1, \exists p \in \mathbb{N}, [\forall q \in \mathbb{N}, q \not= 1, q \not= p, q \nmid p] \land p \mid n$

Which you can also write

$\forall n \in \mathbb{N}, n > 1, \exists p \in \mathbb{N}, \forall q \in \mathbb{N}, q \not= 1, q \not= p, q \nmid p \land p \mid n$

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