definablity of predicates and rigorous proof for definability of a set and a predicate

Question

We have the structure $\mathscr N$ with domain of discourse the set of the natural numbers $\mathbb{N}$, where the language consists of a predicate symbol of arity 3, and a functional symbol of arity 2. The * symbol is interpreted as multiplication of a couple of natural numbers and the predicate symbol b in the following way:

$b^{\mathscr N} \iff x \le z \le y$

Define the following two sets and the two predicates p and q

1. ${\{<n,k>|n,k\in \mathbb{N}, n=k\}}$
2. $\{0\}$
3. $p(x) \iff x=y^3$ for some $y \in \mathbb{N} $
4. $q(x) \iff x=y^2z^2t^2$ for some $y,z,t \in \mathbb{N}$

What I have so far as a solution is the following:

1. $\phi_=(x,y) \equiv b(x,y,x)$
2. $\phi_0(x) \equiv \forall y(\phi_=(*(x,y),x))$
3. $p(x) \equiv \exists y(\phi_=(x,*(*(y,y),y)))$
4. $q(x) \equiv \exists y\exists z\exists t(\phi_=(x,*(*(*(y,y),*(z,z)),*(t,t)))$

My questions are:

• Is my solution correct, formal and rigorous enough?
• How is best to write the solution for such kind of problems formally and rigorously?
• How to define a predicate and what is the difference in defining a predicate and a set in rigorous terms?

Answer 1

Q1,Q2 : You did a good job. I think there's no more rigorous proof. In similar problems, any proof like this will be ok.

Q3. Let language $\mathcal{L}$ be given. Assume n-ary predicate symbol is $P$ in the language. A model/interpretation $\mathcal{M}$ assigns a subset of $D^n$($D$ is domain of a model) to symbol $P$ and denote that subset as $\mathcal{P}^{\mathcal{M}}$

Therefore, $n$-ary predicate (in a model or interpretation) is just a subset of $n$'th power of domain which is assigned by ${\mathcal{M}}$(so predicate is dependent on a model). We denote $(a_1,...,a_n)\in P^{\mathcal{M}}$ as $P^{\mathcal{M}}(a_1,...a_n)$. But this is just mathematical definition.. In many times, we don't distinguish such a set and sentence definable by given model.

(So formally, 'define two predicates p and q' is not right in mathematical terms.'define the two sentences corresponding to p and q' is accurate. But if you don't confuse usual meaning and mathematical definition of predicate, this is somewhat admissible)