What is the transitive closure of the relation $\{(1,5),(5,7),(7,9),(1,7),(1,9),(5,9)\}$ on $X=\{1,5,7,9\}$?
I think this relation is already transitive.
$(1,5),(5,7) \implies (1,7)$ already exist
$(1,5),(5,9) \implies (1,9)$ already exist
$(1,7),(7,9) \implies (1,9)$ already exist
Did I miss something or this relation is already transitive so my answer which is $\{(1,5),(5,7),(7,9),(1,7),(1,9),(5,9)\}$ correct?