understanding of different definition of transitive closure

Question

I'm familiar with the definition of transitive closure, how to find it and its general properties. However, I've run across a different way of defining transitive closure and it is not so intuitive for me. I am hoping someone here can unpack this into words so that this new definition is intuitive for me.

Definition:

Let $X$ be a finite set. Let $B$ be a binary relation on $X$. Then there is a binary relation $T_B$ , called the transitive closure of $B$, defined as:

$\forall {x,y} \in X$ $x {T_B} y$ iff $\exists$ {$x_i$}$_{i=1}^{n} $ $n \geq 1$ where $x_1=x, x_n=y,$$ x_1 B x_2,x_2B x_3.......x_{n-2}B x_{n-1}, x_{n-1} B x_n $

Answer 1

$T_B$ defined like this is characterized by:

• $B\subseteq T_B$
• $T_B$ is transitive
• if $R$ is a transitive relation on $X$ and $B\subseteq R$ then $T_B\subseteq R$

In words $T_B$ is the smallest transitive relation on $X$ that contains $B$ as a subset.

A relation $R$ on a set $X$ is transitive if $xRy\wedge yRz\implies xRz$