I was reading daphne's Probabilistic Graphical Models book and she introduces some notation about sets of random variables that I am confused about (on page 21 section 2.1.3.2).
Before I ask my question let me start with some notation introduced in the book. Let capital non-bold stand for random variables say $X$ is a r.v. Let little non bold stand for the assignment to a random variable say $(X = x)$. Also, let me define captial bold letters as sets of random variables. For example $\textbf{X}, \textbf{Y}, \textbf{Z}$ are three sets of random variables. Let small bold letters denote assigments to these sets $\textbf{x}, \textbf{y}, \textbf{z}$ i.e. it denotes assigments of values to the variables in these sets. Let $Val(\textbf{X})$ be the values that the set of random variables can take.
Finally here is the statement in the book's definition I am having a hard time understanding:
For $\textbf{Y} \subseteq \textbf{X}$, we use $\textbf{x} \langle\textbf{Y} \rangle $ to refer to the assignment within $\textbf{x}$ to the variables in $\textbf{Y}.$
The specific part that confuses me is the word "within". Thats the part I do not understand specifically. I understand that the set of random variables $\textbf{Y}$ has some random variables in common $\textbf{X}$ (or maybe equal), but I am not sure what within means.
Basically, I wish to understand what the notation $\textbf{x} \langle\textbf{Y}\rangle$ means.
As an extention to this notation (and question), the author then extends it by saying the following (which doesn't make sense to me yet since the first part didn't make sense):
For two assigments $\textbf{x}$ (to $\textbf{X}$) and $\textbf{y}$ (to $\textbf{Y}$), we say that $\textbf{x} \sim \textbf{y}$ if they agree on the variables in their intersection, that is, $\textbf{x} \langle \textbf{X} \cap \textbf{Y} \rangle = \textbf{y}\langle \textbf{X} \cap \textbf{Y} \rangle$.