(reflexivity,symmetry,transitivity) relations

Question

Construct a smallest possible relation on the set {a, b, c, d} that is irreflexive, symmetric, and transitive. I thought that the relation would be an empty set, R is symmetric because (a,b) can never belong to it and so the conditional statement is always true. R is transitive because (a,b) and (b,c) can never belong to it and so the conditional statement is always true. do you think that there is anything wrong with my assumption?

Answer 1

Note, that there are in fact, non-empty relations that are not reflexive, but are symmetric and transitive..

A relation $R$ that is not reflexive on a set $\{a, b, c\}$ means simply that one or more of $(a, a), (b, b), (c, c) \notin R$.

So, for example, $R_1 = \{(a, a), (a, b), (b, a), (b, b)\}$ is not reflexive (because $(c, c) \notin R_1$), but it is symmetric and transitive.

However, you stipulated in your question that you were wondering what a relation on a set could be that is irreflexive, symmetric, and transitive would be.

A relation $R_2$ that is irreflexive on a set $A$ is a relation for which there is no element $x\in A$, for which $(x, x) \in R$. So, for example, in our previous example of a non-reflexive relation, we'd have to remove $(a, a), (b, b) \in R_2$. If we remove $(a, a)$ and $(b, b)$ from $R_2$, then it ceases to be transitive, because we'd have $(a, b) \in R_2$, and $(b, a) \in R_2$ but not $(a, a), (b, b)\in R_2$. So we'd need to remove $(a, b), (b, a)$ from $R_2$. Hence we are left with an empty set, $R_2 = \varnothing.$

So the only relation that is irreflexive, symmetric, and transitive, on any set, is in fact, the empty set.