Is a relation on which every element is related with itself alone transitive?

Question

For example,

if A = {1,2,3} and R = {(1,1),(2,2),(3,3)}

is R transitive? If so, then would it be an order relation?

Thanks in advance

Answer 1

Of course it's transitive; we usually call it "equality". And yes, it is also an order relation.

Answer 2

The proof for transitivity is simple:
Let $a,b,c \in A$ and assume that $aRb$ and $bRc$. By definition of $R$, this implies $a=b$ and $b=c$, which implies $a=c$, thus $aRc$, so $R$ is transitive.

Proving antisymmetry and reflexivity is similiar, just remember that $aRb$ iff $a=b$, and substitute that in the definitions.