Expressing statement with predicate logic

Question

Given interpretation ℕ with signature ⟨0, 1; +, ·; =⟩, I need to express statement using predicates: $x$ is not divisible by any prime smaller than $y$

Answer 1

Hint:

Zeroth: How do we write out "w is larger than y"? Let $w > y$ be this formula.

First: How do we write "w divides x"? Let $d(w,x)$ be this formula.

Second: How do we say "w is prime"? - Let $P(w)$ be this formula.

Third: Notice that "$x$ is not divisible by an prime smaller than $y$" is equivalent to saying that "every prime which divides $x$ is greater than or equal to $y$". Hence we can conclude that the formula looks as follows: $$(\forall w) ( [d(w,x) \wedge P(w)] \to [(w > y) \vee (w =y)])$$

I've left it to you to write out $d(w,x)$, $P(w)$, and $w > y$.