Transitive Relationships

Question

I am examining transitive relationships and understand the premise that if $x \rightarrow y \rightarrow z$, then the relation needs to contain $x \rightarrow z$ to be considered transitive. My questions was if I had the relation $A = {(1,2), (2,3), (3,4), (4,5)}$, does including $(1,5)$ make this a transitive relation or would I need also ${(1,3),(1,4),(1,5)}$. If the latter is the case, would I need furthermore to add to the relation for a transitive relation: ${(2,4),(2,5),(3,5)}$?