Is this relation reflexive if it "chains" to itself?

Question

During studying I stumbled upon a thought regarding reflexive relations. I'm familiar that a relation is reflexive if for each element $x$ in a set $S$, $xRx$. (∀x ∈ S: xRx)? Such as something like this $<1,1>,<2,2>$. However is the following deemed a reflexive relation? $$(<1,2>,<2,3>,<3,4>,<4,1>)$$

I'm unsure, as my gut feeling tells me that this relation is not reflexive, yet I am unsure. The addition of $$(<1,1>,<2,2>,<3,3>,<4,4>)$$ would surely make it reflexive if it already was not? Is it already reflexive?

With "chaining" i refer to that the last tuple refers to the start. So something akin to a relation from 4 to 1

Answer 1

No, it is not reflexive. As you wrote, if it has those elements (and it is a binary relation on $\{1,2,3,4\}$), you would have to add the elements $(1,1)$, $(2,2)$, $(3,3)$, and $(4,4)$ to it to get a reflexive binary relation.

Answer 2

To add to what others have said, I think part of why you felt unsure was what you called "chaining" and its relation to another property that a relation can have: transitivity. A relation is transitive if whenever $xRy$ and $yRz$ then $xRz$. A lot of the relations we are used to are transitive. In this case, if you knew $(1,2),(2,3),(3,4),$ and $(4,1)$ were elements in a transitive relation on $\{1,2,3,4\}$ then you could show that the relation would be reflexive.

However, the relation you gave is not transitive (and as others have said, not reflexive).