Union of uncountable set with cardinality c with a countable set

Question

Let $A$ be an uncountable set with cardinality $\mathfrak{c}$ (so it is in bijection with the power set of $\mathbb{N}$) and let $B$ be a countable set (finite or infinite). Intuitively I want to say that the union of $A$ and $B$ has cardinality $\mathfrak{c}$ as well, however I cannot find a way to prove this. How could one go about proving that $|A\cup B| = \mathfrak{c}$?

Answer 1

Suppose $A$ is of size continuum. Then there is a bijection $f\colon A\rightarrow \mathbb{R}$. Let $A_0 = f^{-1}(\mathbb{N})$, a countable subset of $A$. Notice that f restricted to $A \!\setminus\! A_0$ is a bijection between that set and $\mathbb{R} \!\setminus\! \mathbb{N}$, and $f$ restricted to $A_0$ bijects it with $\mathbb{N}$.

Now suppose $B$ is countable and disjoint from $A$. Then $A_0 \cup B$ is countable, and you can use $f$ to define a new bijection $g\colon A_0 \cup B \rightarrow \mathbb{N}$. The definition of $g$ depends on whether $B$ is finite or infinite, but the result in both cases is the same: the union of g with $f \!\restriction\! (A\!\setminus\! A_0)$ is a bijection from $A \cup B = (A\!\setminus\! A_0) \cup A_0 \cup B$ to $\mathbb{R}$.

Can you fill in the details of how to define $g$?