If $L$ is an infinite ($|L|=|\mathbb{N}| $) decidable language, prove that it contains:
a) An infinite subset that is not recognizable.
b) An infinite subset that is recognizable and not decidable.
For the (a) I considered all the subsets of $L$, i.e., $\mathbb{P}(L)$. Since $|\mathbb{P} (L)| = |\mathbb{R}|$, there exists $A \subset L$ that is not recognizable (for there are not enough Turing machines to recognize them all). If $A$ were finite, it would be decidable and hence recognizable, which would be a contradiction. Therefore $A$ is infinite and not recognizable.
For (b) I thought about $B=L\setminus A$. Then $B$ is recognizable because $A$ is not. But I'm not really sure.