The definition of a graph’s transitive reduction

I want to determine the transitive reduction of this graph:


as of now, I only found the first step of doing this: represent the transitive closure of the graph as an adjacency relation, so this is what I did:

 (a,b)  (a,c)  (a,d)  (a,e)  (b,d)  (c,d)  (c,e)  (d,e) 

I’m not sure that this is the correct transitive closure of the graph, and I don’t know how to move forward in determining its transitive reduction.