Let $\mathcal F\subset \mathcal P(\mathbb N)$ be such that $A,B\in \mathcal F\implies A\Delta B$ is finite,where $\Delta$ denotes the symmetric difference .Can the set $\mathcal F$ have the cardinality $c=|\mathbb R|$?
I am stuck with this problem and need some hint.
I am posting an answer by summarizing the discussion in the comment,
Fix $A\in \mathcal F$
Define $\phi:\mathcal F\to \mathcal P(\mathbb N)$ by $\phi(B)=A\Delta B$
$\phi$ is injective as $\phi(B)=\phi(C)=A\Delta B=A\Delta C\implies A\Delta (A\Delta B)=A\Delta (A\Delta C)\implies B=C$.
So,$|\mathcal F|=|\phi(\mathcal F)|$
Now note that $\phi(\mathcal F)\subset \mathcal P_{\text{finite}}(\mathbb N)$,the set of all finite subsets of $\mathbb N$.
$|\mathcal F|\leq \aleph_0$.