Proof involving partitions and equivalence class

[Korisnik]

Homework Statement

Show that every partition of X naturally determines an equivalence relation whose equivalence classes match the subsets from the partition.

Homework Equations

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

The Attempt at a Solution

Let A_i 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∈ A_i.

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

Let x∈A_i. Thus xRx (reflexivity).
Let x, y∈A_i. x, y∈A_i ⇒ y, x ∈A_i ⇔ xRy ⇒ yRx (symmetry).
Let x, y∈A_i and y, z∈A_i then x, z∈A_i ⇔ 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?

 

[Samy_A]

Homework Statement

Show that every partition of X naturally determines an equivalence relation whose equivalence classes match the subsets from the partition.

Homework Equations

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

The Attempt at a Solution

Let A_i 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∈ A_i.

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

Let x∈A_i. Thus xRx (reflexivity).
Let x, y∈A_i. x, y∈A_i ⇒ y, x ∈A_i ⇔ xRy ⇒ yRx (symmetry).
Let x, y∈A_i and y, z∈A_i then x, z∈A_i ⇔ 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?

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

 

[Korisnik]

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