Is the relation $R = \emptyset$ reflexive, symmetric and transitive ? Why?

Question

Can someone help me understand the properties of the relation $R = \emptyset$ ? It looks to me like it's not reflexive, since there is no element related to any element, so the elements are not related to themselves ! Right ? So, does it mean it's irreflexive ?

About symmetry, I guess I understand it's symmetric, since there is not element related to another..

Does the same thing make it asymmetric and antisymmetric ? Or not necessarily ?

Finally, I don't seem to get why it'd be transitive ! If there is no element !

I hope someone has a clear way of understanding this topic. Thanks !

Sorry, I forgot to add that it's a relation on $N^2$ ... therefore, we can say it's reflexive, symmetric, antisymmetric and transitive. It's not irreflexive and not asymmetric ? Right ? Thanks to you all !!

Answer 1

A relation, $R$, is symmetric when for all $x,y$, if $(x,y)\in R$ then $(y,x)\in R$. But here, since $R$ is empty, it has no elements $(x,y)$, so the hypothesis is empty. But the conclusion of an implication is true even with an empty hypothesis. So, $R$ is symmetric.

Same goes for transitivity.

But it is not reflexive, because $(x,x)\notin R, \forall x$, since $R$ is empty. Only in the case that $X=\emptyset$, then again because of empty hypothesis, we have that $R$ is reflexive.

Answer 2

The basic fact here is that an implication "if $P$ then $Q$" is false in one case only: that where $P$ is true and $Q$ is false. For example, to show that the statement $$\hbox{"if $x$ is a bird then $x$ can fly"}$$ is not always true, you have to give an example of something which is a bird but cannot fly.

The statement that $R$ is transitive is that for all $x,y,z$, $$\hbox{if $x\,R\,y$ and $y\,R\,z$ then $x\,R\,z$.}$$ Now if $R$ is the empty relation, then the first part $$\hbox{$x\,R\,y$ and $y\,R\,z$}$$ cannot be true, so the statement as a whole cannot be false. Therefore $R$ is transitive. We often say that $R$ is vacuously transitive.

Answer 3

Whether the empty relation is reflexive or not depends on the set on which you are defining this relation -- you can define the empty relation on any set $X$.

The statement "$R$ is reflexive" says: for each $x\in X$, we have $(x,x)\in R$. This is vacuously true if $X=\emptyset$, and it is false if $X$ is nonempty.

The statement "$R$ is symmetric" says: if $(x,y)\in R$ then $(y,x)\in R$. This is vacuously true, since $(x,y)\notin R$ for all $x,y\in X$.

The statement "$R$ is transitive" says: if $(x,y)\in R$ and $(y,z)\in R$ then $(x,z)\in R$. Similarly to the above, this is vacuously true.

To summarize, $R$ is an equivalence relation if and only if it is defined on the empty set. It fails to be reflexive if it is defined on a nonempty set.