What is the reflexive closure of the empty relation ∅ over a set A?

Question

I understand that R is reflexive if A=∅, and isn't if A is nonempty. But what about the reflexive closure of R?

What is specifically missing from $\mathcal{R}$ that makes it not be reflexive at the moment? If $a\in A$, then you are missing $(a,a)\in\mathcal{R}$, right? How might you notate the set of all things $\mathcal{R}$ is missing then? You can be creative with how you specifically notate it, I'm not sure there is a convenient one-symbol approach here. — JMoravitz, May 20 '16 at 17:28
Reminder, the reflexive closure of $\mathcal{R}$ is the smallest relation $\mathcal{R}'$ such that $\mathcal{R}'$ is reflexive and $\mathcal{R}'\supseteq \mathcal{R}$ — JMoravitz, May 20 '16 at 17:29

score 2 · Accepted Answer · answered May 20 '16 at 17:29

The reflexive closure of some relation $R$ over $A$ is the smallest subset of $A\times A$ that (a) contains $R$ and (b) is reflexive.

In this case $R$ is the empty set, so every subset of $A\times A$ satisfies condition (a). We're left with looking for the smallest subset of $A\times A$ that is a reflexive relation on $A$.

This smallest subset is evidently $\{ \langle a,a\rangle \mid a\in A \}$.

score 0 · Answer 2 · answered Jun 18 '16 at 03:01

I think you should specify the set on which you define the empty relation. Suppose A,B are two sets where A is empty and B is non-empty.Define the empty relation $ R_1$ (respectively $R_2$) on A (respectively on B). Since for every x∈A,(x,x)∈$R_1$, $R_1$ is reflexive. Since there exists x∈B (you can choose any x in B) such that (x,x)∉$R_2$, $R_2$ is not reflexive. The reflexive closure of $R_2$ is $R_2$∪$I_B$ where $I_B$ is the identity relation on B. Hence an empty relation on an empty set and an empty relation on a non-empty set are different.

What is the reflexive closure of the empty relation ∅ over a set A?

2 Answers2