Understanding $PROP$ set in the book Logic and Structure (Van Dalen). Working on the book: Dirk van Dalen. "Logic and Structure (Universitext)" (p. 18)

Definition 1.1.2 The set PROP of propositions is the smallest set X with the properties


$
\begin{array}{rl}
\rm(i)&p_i\in X(i\in N),\bot\in X,\\
\rm(ii)&\varphi,\psi\in X\Rightarrow(\varphi\wedge\psi),(\varphi\vee\psi),(\varphi\to\psi),(\varphi\leftrightarrow\psi)\in X,\\
\rm(iii)&\varphi\in X\Rightarrow(\neg\varphi)\in X.\\
\end{array}
$

I would like to know:

$p_i\in X(i\in N),\bot\in X$


*

*How can I instantiate this statement when verifiyng a string of symbols belongs to PROP ?

*Is the comma an and connective ?

*What is $N$?

*Why is bottom symbol there ?


$((p \land q) \to p)$


*

*How can I show this statement belongs to PROP ?

P.S.: I am already aware of similar questions but they do not address my questions, I think.
 A: 
$p_i\in X(i\in N),\bot\in X$



*

*How can I instantiate this statement when verifiyng a string of symbols belongs to PROP ?


You can instantiate any propositional symbol or $\bot$: $p_1$ is $\in PROP$ according to this clause, $p_{354}$ is $\in PROP$ according to this clause, $\bot$ is $\in PROP$ according to this clause.


*

*Is the comma an and connective ?


Yes.


*

*What is $N$?


It should be $\mathbb{N}$. The $(i \in N)$ just indicates that the $i$'s are running indices to number the propositional symbols.


*

*Why is bottom symbol there ?


Because it is an atomic formula. Unlike the other connectives, it does not take any other formulas to form a new formula, and thus belongs in the base case together with the propositional symbols.

$((p \land q) \to p)$



*

*How can I show this statement belongs to PROP ?


Strictly speaking you can't, because according to the definition introduced up to this point, there exist only indexed propositional symbols starting iwth $p$. But it is customary to use $p, q, r$ in practice; if we incoporate these into clause (i), one can prove by induction on the structure of the formula:

*

*$p \in PROP$ (by (i), with $p_i = p$).

*$q \in PROP$ (by (i), with $p_i = q$).

*Since $p \in PROP$ and $q$ in $PROP$, $(p \land q) \in PROP$ (by (ii), with $\phi = p$ and $\psi = q$).

*Since $(p \land q) \in PROP$ and $p \in PROP$, $((p \land q) \to p) \in PROP$ (by (ii), with  $\phi$ = $(p \land q)$ and $\psi = p$).

