I'm reading the first chapter of "Elements of Set Theory" by Enderton. I found a solution of the book and saw it after doing exercises on my own. But I think there are 3 errors among 7 exercises, which is frequent.
Exercise 1.1. Which of the following become true when $\in$ is inserted in place of the blank? Which become true when $\subseteq$ is inserted?
(a) $\{\emptyset\}$__$\{\emptyset,\{\emptyset\}\}$
(b) ...
(c) ...
(d) ...
(e) ...
Solution. Choices (a) and (d) become true when $\in$ is inserted in place of the blank. Choices (b) and (c) become true when $\subseteq$ is inserted in place of the blank. Choice (e) is not true in either case.
My solution. Choice (a) becomes true when $\in$ or $\subseteq$ is inserted in place of the blank. ...
Assume $V_{a+1} = \mathscr P\,V_a$ and $V_0 = \emptyset$
Exercise 1.5 Define the rank of a set $c$ to be the least $a$ such that $c\subseteq V_a$. Compute the rank of $\{\{\emptyset \}\}$. Compute the rank of $\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset \}\}\}$.
Solution. Observe that $V_{a+1}=\mathscr P\,V_a$. Taking $V_0 = A = \emptyset$, it follows that $$V_1 = \mathscr P\,V_0 = \{\emptyset, \{\emptyset\}\}$$
Hence the rank of $\{\{\emptyset\}\}$ is 1. ...
My solution. $V_1 = \mathscr P\,V_0 = \{\emptyset\}$. $V_2 = \mathscr P\,V_1 = \{\emptyset, \{\emptyset\}\}$ so the rank of $\{\{\emptyset\}\}$ is 2. ...
Exercise 1.7 List all the members of $V_3$. List all the members of $V_4$.
Solution. Without listing them, $V_3$ has 16 members, and $V_4$ has 32 members.
My solution.$V_3$ has 4 elements and $V_4$ has 16 elements.
I think my corrections are all true. So I think a writer of this solution didn't take a lot care in it. But I have no answer to compare without this. When I start proof from axiom and exactly proved theorem, that answer won't be likely to be wrong. But how do I know I'm not mistaken when a process of proving gets long? Is it inevitable being wrong to some degree when studying mathematics at any level?
