Relation $\;\rho\;$ is given on the power set of $\;\mathbb{N}\;$ and is defined with $A\rho B\Leftrightarrow A\cap B\ne\emptyset \land A\cup B=\mathbb{N}\;$.
$\rho\;$ is a symmetric relation. How do I find the smallest equivalence relation that contains $\rho$ and how do I find the largest equivalence relation that is contained in $\rho$?
HINT: The relation $\rho$ isn’t reflexive, so it does not contain any equivalence relation on $\wp(\Bbb N)$. There is exactly one subset of $\wp(\Bbb N)$ on which it is reflexive, and it does contain an equivalence relation on that subset; what is that subset, and what is the equivalence relation in question?
For the first part of the question note that you will definitely have to add a lot of ordered pairs just to get a reflexive relation containing $\rho$. Then you’ll have to worry about transitivity. Note that if $A$ and $B$ are any non-empty subsets of $\Bbb N$, then $A\mathrel{\rho}\Bbb N$ and $B\mathrel{\rho}\Bbb N$, so if $\hat\rho$ is an equivalence relation containing $\rho$, you will have to have $\langle A,B\rangle\in\hat\rho$.