An example in the fundamental theorem of equivalence relations?

Question

I've read about the fundamental theorem of equivalence relations. The idea that an equivalence relation on a set $X$ partitions $X$ is understandable. But the idea that for any partition of $X$ there is an equivalence relation on $X$ is a little weird.

Suppose I partition $\mathbb{N}$ as follows:

$$\{\{\text{primes}>3\},\{\text{even numbers}\},\{1\}\}$$

Then what would be the corresponding equivalence relation?

Answer 1

The equivalence relation would be "belongs to the same set of the partition". Nobody said that it had to be expressed without referring to the partition.

(And where is $9$ in your partition?)

Answer 2

Let $P$ be a partition of set $S$. Then we can define the relation $R=\{ (a, b)\mid \exists T\in P: a\in T \wedge b\in T\}$. Which is basically the relation "belongs to the same set of the partition" (as said by the other answer). This relation can quite easily be proven to be an equivalence relation.

For some intuitive understanding:

Reflexive: an element $x$ of $S$ does of course belong to the same set of the partition as $x$.

Transitive: if $x$ is in the same set of the partition as $y$ and $y$ is in the same set of the partition as $z$ then $x$ and $z$ belong to the same set of the partition.

Symmetry: if $x$ is in the same set of the partition as $y$, then $y$ is in the same set of the partition as $x$.

You should be able to write a more formal proof on your own.