sorry for this really stupid question.
Consider the structure $A=<\mathbb{N},<;\cdot,1>$ (arity type $<2;2,0>$). Give formulas whose interpretations in the model represent the predicates
(a) "$x_2$ is a divisor of $x_1$";
(b) "$x_1$ is prime";
(c) "for every $n$, there is some prime greater than $n$".
Solutions:
So let us begin with (a) and then work our way to (c): (a) First of all I have some question marks: If I am correct, I would like to express $x_1=x_3\cdot x_2$. Is it then correct to write something like this $$\forall x_1\exists (x_2\land x_3)(x_1\doteq f_1(x_3,x_2))?$$
I mean, every natural number can be written as factors, may it be 1 times itself or any other number, every natural number can be written as factors. Therefor i choose to write $\forall x_1$. Then I choose to write that there exists some natural number $x_2$ and some natural number $x_3$ such that this property holds. Is this the correct way to express that $x_2$ is a divisor of $x_1$? Also is $\exists (x_2\land x_3)$ the same as $\exists x_2\land\exists x_3$?
If this is the correct way of saying it, let's head over to (b) where i want to express that $x_1$ is a prime number. I suppose my goal is to use (a) and say that "okey, every divisor of $x_1$ is either 1 or $x_1$". But how would I go on and write that? I guess I could do it something like this $\exists (x_1\land x_2)\text{ ... } (x_1\doteq f_1(f_2,x_2)\lor x_1\doteq f_1(x_1,x_2))$ Where I am not quite sure what I should substitute for $...$ I want somehow to express "for every divisor of $x_1$". Should I do it something like this, to express every divisor of $x_1$? $\forall (x_1=f_1(x_3,x_2))$? If not, what's wrong? Or perhaps everything i wrote is wrong? :) Lastly, for (c), I suppose I have to write it something like this: $\forall n\exists x_1(P(n,\varphi))$ where $\varphi$ is the formula for the prime number?
What am I doing wrong here? What have I mistunderstood? I want to become better at understanding how you can make formulas out of statements and vice versa and really hope someone can help me, thanks :)
A few problems going on. First, notation: "$\exists (x_2\land x_3)$", even as part of a formula, is not well-formed. You'd write "$\exists x_2 \exists x_3$" or "$\exists x_2, x_3$".
Semi-formally, how would you express "x divides y"? Well, x divides y iff there's a $z$ such that $y = x \cdot z$. Can you express that in a formula, $\mathsf{D}(x, y)$?
Similarly, x is prime iff $x > 1$ and for all $v$, if $v$ divides $x$ then $v = 1$ or $v = x$. Now express that in a formula $\mathsf{P}(x)$ which makes use of $\mathsf{D}(x)$.
Similarly for (c).