Given the following proof:
A relation R on a non-empty set S is called sequential if for every element (x,y,z) in SxSxS, at least one of the ordered pairs (x,y) and (y,z) belongs to R. Prove or Disprove: Every symmetric sequential relation on a non-empty set is an equivalence relation.
Would I have to show that the relation R is both reflexive and transitive in order for the proof to be true?
Unless I misunderstand the definition, it seems to me that once the relation is sequential and symmetric, you have that $(x,y) \in R$ for any $x, y \in S$. In other words, $R$ is a total relation: everything is related to everything! (and that makes $R$ trivially an equivalence relation)
Why?
Given that it is sequential, we have that for the element $(x,y,x) \in S \times S \times S$: either $(x,y) \in R$ or $(y,x) \in R$. But given that it is symmetric, if one of these is in $R$, then the other is in $R$ as well.