Image of $\mathbb{N}\setminus\{n\}$ under the map $T$

Question

Let, $\phi:\mathbb{N}\rightarrow\mathbb{N}$ such that, $$\phi(n)=\begin{cases} \frac{n}{2} \text{ when n is even} \\ 3n+1 \text{ when n is odd} \end{cases}$$

Let, $M=\{A\in P(\mathbb{N}):\operatorname{card}(A)\geq 3\}$. Define, $T:M\rightarrow M$ by, $$T(A)=\{\phi(a):a\in A\}$$ Let, $B=\mathbb{N}\setminus\{n\}$ for a fixed $n\in \mathbb{N}$, then clearly $B\in M$. Now my question is :

Is there any $n\in \mathbb{N}$ for which, $T(B)=\mathbb{N}$?

Here, $P(\mathbb{N})$ is the power set of natural numbers.

Answer 1

Sure. Take $n=1$. Then $B=\mathbb{N}\setminus\{1\}$ and $T(B)=\mathbb N$ because, if $n\in\mathbb N$, then $2n\in B$ and $\phi(2n)=n$.