I have some confusion related to definitions in graph theory in Rosen's book.
Firstly, I wonder if an edge, for example u connected to v, is expressed as (u,v) or {u,v}.
Secondly, Rosen mentions the definition that a digraph is weakly connected if there is a path between any two vertices in its underlying undirected graph. As a result, is it analogous to there is a path between any two vertices in the digraph, regardless of direction, e.g, there is a path from u to v (maybe not from v to u).
Thank you so much and I would be extremely grateful.
Not having the book at hand, but for digraphs it is usually $(u,v)$ (ordered pair). Else you couldn't know what's the source and what's the target of the edge (because $(u,v)\neq (v,u)$ but $\{u,v\}=\{v,u\}$). Because that doesn't matter in simple graphs, there $\{u,v\}$ is used.
I'm unsure what's your question regarding the second part. A digraph is weakly connected if it's underlying undirected graph is connected, but that does not imply that for every two different vertices u,v there is a (directed) path from u to v or v to u. Take for example $G=(\{-1,0,1\}, \{(-1,0), (1,0)\})$. $G$ is weakly connected, but there is no directed path from -1 to 1 or from 1 to -1.
On a site note, graph theory is probably the branch of mathematics where are the most notational differences between authors :)