Suppose that S is a nonempty set and C is a partition of S. Show that there is a unique equivalence relation ∼ on S with equivalence classes being the sets in C.
By unique I mean there exist a few equivalence classes such that they are different.
My thinking process is define a equivalence relation, but I am not sure how to argue that equivalence classes are different from each other.
Fix a set $X$. Let $P$ be the set of partitions of $X$. Let $C$ be the set of equivalence relations on $X$. Define $f : P \to C$ by $f(p) = \sim_p$, where $\sim_p$ is defined by $x \sim_p y$ if and only if there exists an $a \in p$ such that $x,y \in a $. Define $g : C \to P$ by $g(\sim) = p_\sim$ where $$ p_\sim = \{C_x \subset X \ \mid \ x \in X \text{ and } (y \in C_x \iff x \sim y )\} $$ Can you show that $g = f^{-1}$? If so, this establishes a bijection between the set of partitions of a set and the set of equivalence relations (hence equivalence classes) on a set.