Function $f\colon 2^{\mathbb{N}}\to 2^{\mathbb{N}}$ preserving intersections and mapping sets to sets which differs only by finite number of elements

Define on $2^{\mathbb{N}}$ equivalence relation $$X\sim Y\Leftrightarrow \text{Card}((X\setminus Y)\cup(Y\setminus X))<\aleph_0$$ Is there exist a function $f\colon 2^{\mathbb{N}}\to 2^{\mathbb{N}}$ such that $$f(X)\sim X$$ $$X\sim Y \Rightarrow f(X)=f(Y)$$ $$f(X\cap Y)=f(X)\cap f(Y)$$

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No. Think about what would happen with an uncountable almost disjoint family. – user83827 Nov 16 '11 at 23:21
Sets ${X_\alpha}$ of such a family will belong to different equialence classes. From properties 1 and 2 we see that it implies $f(X_\alpha)\cap f(X_\beta)=\varnothing$. If I understood almost disjoint mean $\text{Card}(X_\alpha\Delta X_\beta)\geq\aleph_0$. Hence $f(X_\alpha\cap X_\beta)\nsim\varnothing=f(X_\alpha)\cap f(x_\beta)$. Right? – Norbert Nov 16 '11 at 23:33
Not quite. Two sets are almost disjoint if their intersection is finite. Your properties imply that $f(X_\alpha)\cap f(X_\beta)=f(X_\alpha\cap X_\beta)=f(\varnothing)$ whenever $\alpha\ne\beta$. Now look at the sets $f(X_\alpha)\setminus f(\varnothing)$. – Brian M. Scott Nov 16 '11 at 23:38
Thanks, now the rest is clear. – Norbert Nov 16 '11 at 23:43
Why don’t you go ahead and write up the answer yourself, so that the question doesn’t stay on the Unanswered list; this is not only okay, it’s explicitly encouraged. – Brian M. Scott Nov 17 '11 at 0:14

Let $\{X_\alpha : \alpha\in\mathcal{A}\}\subset\mathbb{N}$ be an uncountable family of sets such that $$\alpha,\beta\in\mathcal{A},\quad\alpha\neq\beta\Rightarrow \text{Card}(X_\alpha\cap X_\beta)<\aleph_0$$ Such a family does exist. Indeed for each irrational number $x\in\mathbb{I}$ consider sequence of rational numbers $\{x_n\}_{n=1}^{\infty}\subset\mathbb{Q}$ tending to $x$. Let $\varphi(x)=\{x_n:n\in\mathbb{N}\}$ be the set of this rational numbers. Obviously for $x,y\in\mathbb{I}$ such that $x\neq y$ we have $\text{Card}(\varphi(x)\cap\varphi(y))<\aleph_0$. Also obviously for all $x\in\mathbb{I}$ we have $\text{Card}(\varphi(x))=\aleph_0$. Let $i\colon 2^\mathbb{Q}\to 2^\mathbb{N}$ be some bijection between $2^\mathbb{Q}$ and $2^\mathbb{N}$ then we may take by definition $\{X_\alpha : \alpha\in\mathcal{A}\}=\{i(\varphi(x)):x\in\mathbb{I}\}$. This will be desired family.
Let $\alpha,\beta\in\mathcal{A},\alpha\neq\beta$. Then $X_\alpha\cap X_\beta\sim\varnothing$. And from the second and third properties we obtain $f(X_\alpha)\cap f(X_\beta)=f(X_\alpha\cap X_\beta)=f(\varnothing)$.
Now for each $\alpha\in\mathcal{A}$ consider $Y_\alpha=f(X_\alpha)\setminus f(\varnothing)$. By construction $X_\alpha$ is infinite, so does $f(X_\alpha)$, and as the consequence $Y_\alpha\neq\varnothing$. Now for all $\alpha,\beta\in\mathcal{A},\alpha\neq\beta$ we have $$Y_\alpha\cap Y_\beta=f(X_\alpha\cap X_\beta)\setminus f(\varnothing)=\varnothing$$ Thus we built an uncountable family of disjoint subsets $\{Y_\alpha : \alpha\in\mathcal{A}\}$ in countable set $\mathbb{N}$. Contradiction, hence such a function doesn't exist.
I've corrected Let $i\colon 2^\mathbb{Q}\to 2^\mathbb{N}$ be some bijection between $\mathbb{Q}$ and $\mathbb{N}$ to $i\colon \mathbb{Q}\to \mathbb{N}$.\\ I guess you should also mention that $Y_\alpha$'s are non-empty to get a contradiction. – Martin Sleziak Nov 17 '11 at 13:19
You corrected my typo to another typo. In first redaction a meant bijection between $2^{\mathbb{Q}}$ and $2^\mathbb{N}$. I meant exactly such a bijection because $\varphi$ maps $\mathbb{I}$ into $2^\mathbb{N}$. So I will edit my post again. – Norbert Nov 17 '11 at 13:41