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**wgunther** Firstly, $\displaystyle a\cup Y = a\cup Y$ so $\displaystyle aRa$ for all $\displaystyle a\in\wp(X)$.

Suppose $\displaystyle aRb$ and $\displaystyle bRc$. Then $\displaystyle a\cup Y=b\cup Y=c\cup Y$. So, by symmetry and transitivity of usual set equality, $\displaystyle bRa$ and $\displaystyle aRc$ So it's an equivalence relation.

For b, what is the equivalence class of $\displaystyle A=\{\,1,2\,\}$ is asking what things are R-related to A. If B is R-related to A, then $\displaystyle A\cup Y=B\cup Y$. Then clearly, as 1 and 2 aren't members of Y, we can conclude that $\displaystyle \{\,1,2\,\}\subseteq B$ and furthermore, 3 or 4 could be on B (that wouldn't effect anything since they would be in their after unioned with Y anyway). So it's equivalence class is

$\displaystyle \{\,\{\,1,2\,\},\{\,1,2,3\,\},\{\,1,2,4\,\},\{\,1, 2,3,4\,\}\,\}$