currently trying to do the following task:
formalize statements about natural numbers using the following signature:
$ \Sigma = (F_\Sigma, P_\Sigma, \alpha_\Sigma) =$ $ (\{+, \ast\}, \{=, <, \text{{get}}\}, \alpha_\Sigma) $
$\alpha_\Sigma(+) = \alpha_\Sigma(\ast) = \alpha_\Sigma(=) = \alpha_\Sigma(<) = 2$
$\alpha_\Sigma(\text{{get}}) = 4$
Assume evaluations over the natural numbers universe ($U = \mathbb{N}$) with symbols $+, \ast, =, <$ interpreted conventionally.
The predicate $\text{get}$ has a special function: for each $n \in \mathbb{N}$, there exists a bijection $\text{map}_n : \mathbb{N}^n \rightarrow \mathbb{N}$. The predicate $\text{get}(z, n, j, y)$ evaluates to true if $x_i$ (where $i \neq j$) exists such that $z = \text{map}_n(x_1, \ldots, x_n)$ with $x_j = y$. With $\text{get}$, statements about sequences of arbitrary finite length, each represented by a single $z \in \mathbb{N}$, can be made. Only interpretations $(D, I)$ over $\Sigma$ with symbols interpreted as described are considered for formula evaluation.
Provide a predicate logic formula over $\Sigma$ for each scenario below (with $X, B, $ and $N$ as free variables). Use Infix notation for $+, \ast, =, <$, and $x \leq y$ as syntactic sugar for $(x < y \vee x = y)$.
(a) $X$ is a prime number:
(b) The prime factorization of $X$ includes $B$:
(c) $X$ is equal to $B$ raised to the power of $N$ ($N > 0$):
(d) The prime factorization of $X$ consists of $N$ numbers and includes $B$: For this subtask, you are allowed to use the $\text{{prim}}(y)$ predicate, which is true when $y$ is a prime number (this can be constructed using your solution from Task 1).
so far i've looked up the standart definition of prime numbers in predicate logic, which used implications. As this signature doesnt use implication as a function symbol that wouldnt work. Would appreciate any input on how to use the get function that describes sequences to describe prime numbers.
Thanks and best regards