i was given the solution of a question on Enderton'elements of set theory as below:enter image description here
it turns out i cant very understand why partitions has to do anythings to Equivalence relation at all after hour of wondering and searching for the definition. thank you very much
Generally, when two sets are the same size, that suggests that there is a bijection between them.
Given a partition, there is an obvious equivalence relation of "two elements are equivalent if they are in the same subset". This satisfies all the requirements of an equivalence relation. Obviously all elements are in the same subset as themselves, so this relation is reflexive. If $x$ is in the same subset as $y$, then $y$ is in the same subset as $x$, so it is symmetric. If $x$ and $y$ are in the same subset, and $y$ and $z$ are in the same subset, then $x$ and $z$ are in the same subset, so it is transitive (since all the subsets are disjoint, whatever subset we found $x$ and $y$ has to be the same subset as the one we found $y$ and $z$ in).
So that is a function from partitions to equivalence relations. And it has an inverse: given an equivalence relation, construct a partition such that two elements are in the same subset if and only if they are equivalent.
So this is a bijection between partitions and equivalence relations.
