Some set theory, inductive sets

Question

1. If $X$ is inductive, then the set $\{x\in X\mid x\text{ is transitive}\}$ is inductive. Hence every $n\in\mathbb{N}$ is transitive.

2. If $X$ is inductive, then the set $\{x\in X\mid x\text{ is transitive and }x\notin x\}$ is inductive. Hence, $n\notin n$ and $n\neq n+1$ for each $n\in\mathbb{N}$.

Answer 1

Please ask questions, don't just quote your assingment/problem. It's better if you specify what you have tried or where you are stuck.

To prove that a set $S$ is inductive, you need to prove that:

1. $\emptyset\in S$; and
2. If $a\in S$, then $a\cup\{a\} \in S$.

Conversely, if $S$ is inductive, then 1 and 2 will hold.

So, for your first problem: to show the set you are given is inductive, you must show that $\emptyset$ is in the set. That is, you need to show that $\emptyset \in X$ and that $\emptyset$ is transitive. Since $X$ is assumed inductive, then $\emptyset\in X$. And $\emptyset$ is transitive, so $\emptyset$ is in the set.

Then, assume $a$ is in the set; you need to show that $a\cup\{a\}$ is in the set. That is, assume that $a$ is an element of $X$ and is transitive; use that to show that $a\cup \{a\}$ is in $X$ and is transitive. That will show your set is inductive.

As for the final clause: $\mathbb{N}$ is inductive, so that means that $\{n\in\mathbb{N}\mid n\text{ is transitive}\}$ is an inductive set by what you will have proven by this point. It is also a subset of $\mathbb{N}$. What subsets of $\mathbb{N}$ are inductive?

Part 2 is proven the same way: show $\emptyset$ is in the given set, then show that if $a$ is in the set, then so is $a\cup\{a\}$. The first part of the "Hence" is immediate from the first clause of $2$. for the second part of the "Hence", let me ask you a question: is $n\in n+1$?