I'm reading a book on Discrete Math and came upon this Definition in the Chapter of Graphs which I can't understand. Can anyone help me understand its meaning?
Definition: The set of all neighbours of a vertex $v$ of $ G = (V , E) $ denoted by $ N(v) $, is called the neighbourhood of $v$. If $A$ is a subset of $V$, we denote by $N(A)$ the set of all vertices in $G$ that are adjacent to at least one vertex in $A$. So, $N(A) = \bigcup_{v\in A} N(v)$.
On
If you understand the notion of the neighborhood in graphs, it simply means that $v'\in N(v)$ iff we can go from $v$ to $v'$ in one go, over some of the edges of the graph. Now, we want to have a similar notion not for a single vertex $v$, but for a set of them $A$. We say that $v'\in N(A)$ iff we can go from $A$ (i.e. from some point in $A$) to $v'$ in one go, over some of the edges of the graph. That is, $v'$ is a neighbor of the set $A$ if it is a neighbor of some elemnt $v\in A$. That's why $N(A)$ is natural to define as $N(a) = \bigcup_{v\in A}N(v)$.
Two vertices $u,v\in V$ are said to be adjacent if $uv\in E$, and a vertex $u$ is a "neighbor" of a vertex $v$ if $u$ and $v$ are adjacent. So $N(v)=\{u\in V: uv\in E\}$ and if $A\subset V$ then $N(A)=\bigcup_{v\in A}N(v)$. Is that more clear?