Finite and infinite sets, cardinality question

Question

Suppose there are infinite sets $A$, $B$ and $C$ such that $$|A| = |B| = |C| = |\mathbb{N}|\\ |D| = |\mathbb{R}|$$ and the finite set $E$

Give an example for the following (using the sets above). In case it's not possible, show why.

1. $(A \setminus D = B) \wedge (A \cap D = C)$
2. $\mathcal P(E) \setminus A = B $
3. $|D| = |E|^{|A|}$

This is an exam type of exercise i couldn't answer it, if there is a soul that can help, I'll appreciate it.

Answer 1

Here are some hints:

1. Start with the case of $D'\cap A=\varnothing$, and find a suitable subset $C$. For example $D'$ can be the irrational numbers, $A$ the rational numbers, and $C$ the natural numbers.
2. This is impossible $B$ is infinite but $E$ is finite. The power set of a finite set is finite, and $\mathcal P(E)\setminus A$ is a subset of $\mathcal P(E)$.
3. Remember that $\Bbb R$ and $\mathcal P(\Bbb N)$ are equipotent.