Denote $\Bbb N_+ = \Bbb N \setminus\{0\}$. The Boolean ring $\mathscr{P}(\Bbb N)$ has an ideal $\mathscr{P}(\Bbb N_+)$. Find the elements of $\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)$ and compute the multiplication and addition tables.
Since this is asking me to compute the tables I feel like there should be a finite number of elements in $\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)$, but I have that $$\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)=\{A \triangle \mathscr{P}(\Bbb N_+) \mid A \in \mathscr{P}(\Bbb N)\} = \{\{A \triangle B \mid B \in \mathscr{P}(\Bbb N_+)\} \mid A \in \mathscr{P}(\Bbb N)\}$$ and for some reason this collection doesn't seem to be finite?
The sets $\Bbb N$ and $\Bbb N_+$ only differ by one element 0, but what about $\mathscr{P}(\Bbb N)$ and $\mathscr{P}(\Bbb N_+)$ it seems that the cardinality of $\mathscr{P}(\Bbb N)$ is $2^{\aleph_0}$ so is the cardinality of $\mathscr{P}(\Bbb N_+)$ equal to $2^{\aleph_0 -1}$ does the set $\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)$ contain only one element? I'm getting really confused here.
Hint: Prove that $\{A\triangle B:B\in P(\Bbb N_+)\}=P(\Bbb N_+)$ if $0\notin A$ and it's $=\{\{0\}\cup S:S\in P(\Bbb N_+)\}$ if $0\in A$.