elementary set theory - Size of infinite sets equality #$A$=#($Acup B$) and counterexample for $Acap Bneemptyset$

Let $A$ be an infinite set and $B$ a countable set and let them be disjoint.

Then there exists an injection $j:\Bbb N\hookrightarrow{} A$.

We can call its image $C:=j(\Bbb N)\subset A$. Then we have that $\Bbb N\cong C$ ($C$ is equipotent $i.e.$ there is a bijection between the two sets).

Since $B$ is countable and $C$ is countable and infinite, $B\cup C$ is countable and infinite.

So $B\cup C\cong \Bbb N\cong C\implies\exists \varphi:B\cup C\xrightarrow{\text{bij}} C$

Now I have two questions:

1. Why is the function defined as $$\psi:A\cup B\rightarrow A\ , x\rightarrow\begin{cases}\varphi(x)\ \text{if}\ x\in C\cup B\\ x\ \text{if}\ x\in A\end{cases}$$ bijective?




2. Is there an example of $A,B$ that are not disjoint and $A\cup B \ncong A$?