Consider a relation $R=\{(a,a): a\in A\}$ where $A=\{1,2,3\}$.
i.e $R=\{(1,1),(2,2),(3,3)\}$

This clearly reflexive but is it necessary that all such relations are necessary to be symmetric and transitive?

In other words:
By definition $R$ is said to be symmetric if $(a,b)∈R \implies (b,a)∈R$. $\longleftarrow$*even if $a=b$?*
and $R$ is said to be transitive if $(a,b)\in R$ and $(b,c)\in R \implies (a,c)\in R$. $\longleftarrow$*even if $a=b=c$?*


1 Answer 1


You are correct. There is nothing in the definition of a transitive relation that prohibits it from only holding on pairs of equal elements. It just says:

take any $2$ elements from the set. Is $(a,b)\in R$? Then $(b,a)$ must be in $R$. The rule says nothing for a pair $(a,b)$ which is not in $R$, and since id demands nothing, it is by default satisfied.

Notice that the only rule preventing an empty relation from being an equivalence relation is really the reflexive demand (the relation $\emptyset$ is transitive and symmetric, but not reflexive), so your "diagonal" relation of $$R_m=\{(x,x)|x\in A\}$$ is in fact the "smallest" equivalence relation on $A$ (it is a subset of all other equivalence relations). However, you usually do not think of $R_m$ as an equivalence relation, since it is a much more known relation of equality: $(x,y)\in R_m \Leftrightarrow x=y$.


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