I came across this in a university textbook.
Let $A$ be a nonempty set and $S$ is its partition. We define a relation $R$ as $$R = \{(a,b) \in A \times A ~|~ \exists M \in S \land a \in M \land b \in M\}$$
Then $R$ is equivalence relation on $A$ and $S$ is its partition.
Then in the textbook follows a proof of the statement, of which the first sentence is
Because all sets in $S$ are pairwise disjoint, the definition of relation $R$ is correct.
My question is: why wouldn't the definition of $R$ be correct if the sets of $S$ weren't disjoint? Of course that then $S$ wouldn't be a proper partition of $A$, but does it somehow affect the correctness of definition of $R$?
The definition of R is correct no matter what S may be. Even if S were empty, R would simply be empty. The significant point of the discussion however, is that R is an equivalence relation for A iff S is a partition of A.