3

Given a non-empty set, A, and an empty relation, R, on that set A, can it be the case that the relation R is an equivalence class?

Transitivity. (a,b) in R, (b,c) in R ===> (c,a) in R. This is trivially true because there is no a, b, c, in R, so the implication is true by way of the antecedent being false.

Symmetry. (a,b) in R ==> (b,a) in R. Again, trivially, the implication is true because the antecedent is false.

But what about reflexivity. If a in R, then a in R. But no a in R -- is this true or false?

compguy24
  • 421
  • I believe reflexivity is equivalent to the condition that $\forall a\in A (\exists b\in A, (a,b) \in R) \implies (a,a) \in R$ This is vacuously true. – ruler501 Mar 27 '14 at 22:18
  • 4
    Reflexivity is $\bigl(\forall a\bigr)\bigl((a,a)\in R\bigr)$. – Daniel Fischer Mar 27 '14 at 22:19
  • 1
    Agreeing with @DanielFischer's definition, in which case the empty relation is not an equivalence relation. If you take ruler501's definition, then it is an equivalence relation, but I don't think this definition is standard. Either way, "If a is in R, then a is in R", doesn't make sense as a definition of reflexivity for two reasons: first, it seems like you mean a to be an element of A, in which case it can't also be an element of $R$. And second, the statement "if x, then x" is always true. Always. It doesn't need a name like "reflexivity". – Abram Lipman Mar 27 '14 at 22:25
  • I'm not sure if my definition is right or not it just seemed the closest to what he had said that made logical sense. – ruler501 Mar 27 '14 at 22:32

1 Answers1

6

An equivalence relation on a non empty set can't be empty, because it's reflexive. So, for any $a\in A$, you have $(a,a)\in R$. Now there is some $a\in A$.

From a slightly different point of view, an equivalence relation on $A$ always contains the identity relation $$ \Delta_A=\{(a,a):a\in A\} $$ which is empty if and only if $A$ is empty.

It's true that the empty relation is transitive and symmetric (also antisymmetric, by the way) on every set.

Note that the main purpose of equivalence relations is to partition the set into pairwise disjoint subsets: $$ [a]_R=\{x\in A: (a,x)\in R\} $$ and so reflexivity (hence non emptyness of $R$) is essential to have $a\in[a]_R$. With the empty relation you can't have a partition defined this way.

egreg
  • 238,574