# Proof involving partitions and equivalence class

1. Feb 3, 2016

### Korisnik

1. The problem statement, all variables and given/known data
Show that every partition of X naturally determines an equivalence relation whose equivalence classes match the subsets from the partition.

2. Relevant equations
( 1 ) we know that equivalence sets on X can either be disjoint or equal

3. The attempt at a solution
Let Ai be a partition of X, i ∈ I, where I is an enumeration set, R is a relation on X. Let x,y∈X, xRy ⇔ ∃i∈I : x, y∈ Ai.

Since X is partitioned into sets, we choose Ai by taking an arbitrary number i∈I, Ai isn't empty.

Let x∈Ai. Thus xRx (reflexivity).
Let x, y∈Ai. x, y∈Ai ⇒ y, x ∈Ai ⇔ xRy ⇒ yRx (symmetry).
Let x, y∈Ai and y, z∈Ai then x, z∈Ai ⇔ xRy ∧ yRz ⇒ xRz (transitivity) (1).

Thus R is an equivalence relation.

The part so far is pretty clear. What I actually have to prove is the bolded part, and I'm not sure what is it that I have to prove. So far I've done this:

By definition of an equivalence class, each such equivalence relation (such that every pair of two elements of the same subset are in that relation) determines an equivalence class on each different subset.

Which essentially means: it's obvious from the definition... and I'm not satisfied with it. What should I prove, i.e. how do you formalize “equivalence classes are equal to subsets in a partition set”? How to start?

2. Feb 3, 2016

### Samy_A

You don't have to start, you are already done.
"Let x,y∈X, xRy ⇔ ∃i∈I : x, y∈ Ai" is the bolded text in symbols, and you have proved that this defines an equivalence relation.

3. Feb 3, 2016

### Korisnik

Okay, I guess you're right. I should leave it alone.