Understanding Equivalence Relations and the Role of the Empty Set

[Bleys]

Given any set A, a relation on A is a subset of AxA. Then isn't the empty set a relation also? Doesn't that make it an equivalence relation, vacuously, as well?
I'm asking because in a book there's a problem stating: show there are exactly 5 equivalence relations on a set with 3 elements. I get the obvious
{(1,1), (2,2), (3,3)}
{(1,1), (2,2), (3,3), (1,2), (2,1)}
{(1,1), (2,2), (3,3), (1,3), (3,1)}
{(1,1), (2,2), (3,3), (2,3), (3,2)}
{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)} = AxA
But I think the empty set should also be included, because for example in {(1,1), (2,2), (3,3)}, symmetry and transitivity are both trivially satisfied, just as they would be in the empty set.

But I know equivalence relations correspond to partitions of the set. Then the partitions would be
{1} {2} {3}
{1,2} {3}
{1,3} {2}
{2,3} {1}
{1,2,3}
And the empty set doesn't partition A, so what should it be?
How is the empty set regarded with respect to (equivalence) relations?

 

[tiny-tim]

Hi Bleys!

Doesn't an equivalence relation have to be reflexive?

So (as a subset of AxA), it must at least contain every (a,a)

(which the empty set doesn't)

 

[JSuarez]

The empty set is indeed a relation on A (with indefinite arity), but not an equivalence one, except in the case where A is also empty.

The catch here is that reflexivity is not (like transitivity or symmetry) expressed as an implication; its formal statement is just:

$$\forall x\left(xRx\right)$$

Which fails unless A itself is empty.

 

[Tac-Tics]

Given any set A, a relation on A is a subset of AxA. Then isn't the empty set a relation also? Doesn't that make it an equivalence relation, vacuously, as well?

Kind of.

It's at this point you realize why vacuous definitions are vacuous -- they don't really matter. They have no substance. No one really cares about them because they are totally definition-driven.

I could easily write my own textbook saying a (binary) relation on a set A is a subset of A x A which has at least one member. My definition is virtually identical to every other math book. All of their proofs will work under my definition because no one writes proofs for vacuous theorems!

The only difference is I could (if I wanted to) remove all restrictions when a proof requires a non-empty relation. In fact, the equivalent relation-partition theorem imposes this non-empty relation requirement. Go back and check your particular text. Either your text defines relations to be non-empty or the theorem only applies to non-empty relations (or your book has an error).

 

[blue_raver22]

I can provide a response to this question by first clarifying the definition of an equivalence relation. An equivalence relation is a relation on a set that is reflexive, symmetric, and transitive. This means that for any element in the set, it is related to itself (reflexive), if two elements are related to each other, then they are also related in the reverse direction (symmetric), and if two elements are related to each other and one of them is related to a third element, then the first element is also related to the third element (transitive).

Now, to address the question about the empty set being an equivalence relation, I would say that it depends on the definition of a relation. While some definitions may include the empty set as a relation, others may not. However, even if we do consider the empty set as a relation, it does not necessarily make it an equivalence relation.

For example, let's consider the set A = {1,2,3} and the empty set as a relation on A. The empty set would be represented as {} and it is indeed a subset of AxA. However, it is not reflexive, symmetric, or transitive, as there are no elements in the set to relate to each other. Therefore, the empty set is not an equivalence relation.

In terms of the problem of finding the number of equivalence relations on a set with 3 elements, the empty set would not be included as it does not meet the criteria of an equivalence relation. The 5 equivalence relations listed in the question are the only ones that satisfy the definition.

In terms of partitions, the empty set would not be considered a partition as it does not divide the set into non-empty subsets. Therefore, it is not included in the list of partitions that correspond to the 5 equivalence relations.

In conclusion, while the empty set can be considered a relation, it does not meet the criteria to be considered an equivalence relation. It is not included in the list of equivalence relations or partitions in this context.