TITLE: Is an empty set reflexive? Symmetric? Transitive?
QUESTION [5 upvotes]: Suppose
$$A\neq\emptyset$$
Since, $$\emptyset\subseteq A\times A$$
the set $$R=\emptyset$$ is a relation on A.
Is $R$ reflexive? symmetric? transitive?
I remember hearing something can be "vacuously" true. So the empty set would be reflexive, symmetric and transitive because it doesn't meet the definition?
So there is no $(x,x)$ that can exist in $R$ therefore vacuously reflexive.
There is no $(x,y)$ that can exist in $R$ therefore vacuously symmetric.
There is no $(x,y)$ that can exist in $R$ therefore vacuously transitive.
Is my reasoning correct here?

REPLY [6 votes]: I think its transitive automatically because the relation only has the empty set but I'm not sure.

The term is "vacuously".
A relation is transitive if $\forall x\forall y\forall z \Big((x,y)\in R\wedge (y,z)\in R \to (x,z)\in R\Big)$.
This is vacuously true because you cannot find any counterexamples, since the relation is empty.  (The implication is never falsifiable)


So there is no $(x,x)$ that can exist in $R$ therefore vacuously reflexive

No, the set $A$ is not empty, so $\forall x( x\in A\to (x,x)\in R)$ is not a vacuous truth; it is in fact fallacious.
However, the definition for irreflexive is $\forall x( x\in A\to (x,x)\notin R)$, so that is true, although not vacuously so.

There is no $(x,y)$ that can exist in $R$ therefore vacuously symmetric

Yes, symmetry requires $\forall x \forall y \Big((x,y)\in R \to (y,x)\in R\Big)$.
Now, what about antisymmetry, and asymmetry?

There is no $(x,y)$ that can exist in $R$ therefore vacuously transitive

Done