# Is this relation reflexive if it “chains” to itself?

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

• No to the first question, yes to the second. Trust your gut in this case ;) – Don Thousand Mar 29 '19 at 18:55

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.
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.