Relations/Partitions/Exercise

Let ${}M$ be a set, and let ${}P\subseteq {\mathfrak {P}}\,(M)$ denote a partition. Show that ${}P$ induces via

$x\sim y{\text{ if there exists an }}A\in P{\text{ such that }}x\in A{\text{ and }}y\in A,$

an equivalence relation on ${}M$. Compute this relation for the partition ${}\{\{1\},\{2,3,4\},\{5,6\},\{7\}\}$ of the set ${}\{1,2,3,4,5,6,7\}$.

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