Does a reflexive element constitute asymmetry and anti-symmetry?

Question

I'm studying properties of relations and there is one area that i'm kind of unsure about regarding the properties of asymmetry and anti-symmetry.

Suppose $R = \{(1,2),(3,4),(2,2)\}$

It would follow that $R$ is:

Not reflexive, Not irreflexive, Not symmetric,

I would say it is not asymmetric and not antisymmetric also, but I get hung up on the $(2,2)$ element.

Does $(2,2)$, or any reflexive ordered pair, count as $a = 2$ and $b = 2$, such that $a R b$ and $b R a$ ?

Answer 1

Not reflexive Correct. Not every element is related to itself.

Not irreflexive Correct. Since we have $(2, 2)$. Irreflexivity requires that no element should be related to itself.

Not symmetric Correct. We have $(3, 4)$ but not $(4, 3)$.

Antisymmetric Correct, the only instance where $(a, b) \in R$ and $(b, a) \in R$ is $a = b$ (the pair$(2, 2))$

Not transitive ?

It is transitive because $(1, 2)$ and $(2, 2)$.

Not that this relation is defined on a set, which has a unique element 2, it seems to be the point of confusion from the comments.

Answer 2

Taken any $a$, $b$ and $c$, we have:

$(a,b)$ and $(b,a)$ $\in R$ $\Rightarrow$ $a = b = 2$ because there are no other such couples in $R$. This shows $R$ to be anti-symmetric.

$(a,b)$ and $(b,c)$ $\in R$ $\Rightarrow$ $a = 1$, $b=c=2$ or $a=b=c=2$. In both cases: $(a,c) \in$ $R$. This is why $R$ is transitive.