Again, a question based on "An Invitation to Applied Category Theory". This time, exercise 1.15 on page 10.
Here we have an equivalence relation $\sim $.
Under $ \sim $ we form a partition on the set $A$. Say that a subset $X \subseteq A$ is ($ \sim$) -closed if for every $x \in X$ and $x' \sim x$ we have $x' \in X$. We say that a subset $X \subseteq A$ is ($ \sim$)-connected if it is nonempty and $x \sim y $ for every $x,y \in X$.
We are then asked to show that $A=\bigcup_{p \in P}A_p$
At the start of the given proof we then have "Let $X:= \{a' \in A | a' \sim a \}$." From here it shows that the subset is closed and connected. But what if there was a subset $Z := b'$ and the relation did not apply? I fail to see how saying there is a subset where the relation does apply rules out there being a subset where the relation does not apply. Could someone explain what I have missed here?