The idea of sets being shattered comes up a lot in statistical learning theory with applications to VC dimension. Before learning about this, I am trying to understand the following definition.
Let $F=\left \{ x_{1}, x_{2},...,x_{n} \right \}$ be a finite set. Let $G$ be a subset of $F$. We say that a class of sets $\mathcal{A}$ $\textbf{picks out}$ $G$ if $$A \cap F = G$$ for some set $A \in \mathcal{A}$.
The author then gives the following example:
Consider the class of sets $\mathcal{A} =\left \{ (a,b): a \leq b \right \}$. Suppose that $F=\left \{ 1,2,7,8,9 \right \}$ and $G = \left \{ 2,7 \right \}.$ Then $\mathcal{A}$ picks out $G$ since $A \cap F = G$, if we take $A= \left \{ 1.5,7.5 \right \}$ for example.
Clearly, looking at this example, $$A \cap F = \emptyset \neq G.$$ So, what am I missing?
Thanks in advance!
Note the definition of each element of the class of sets is $\left(a,b\right)$ such that $a \le b$, for some given $a$ and $b$. As such, $A = \left\{1.5,7.5\right\}$ is the set of all real numbers between $a = 1.5$ and $b = 7.5$, which of course includes $2$ and $7$, but no other element of $F$, so it's intersection with $F$ is just the elements of $G$, as stated.
Update: I don't really know much about set theory, but I would have thought a better way to express $A$ would be something like $A = \left(1.5,7.5\right)$ (as bof suggests in a question comment), or perhaps something like $A = \left\{x : 1.5 \lt x \lt 7.5\right\}$. Otherwise, as currently expressed, it seems to be a set with just $2$ elements, i.e., $1.5$ and $7.5$, which is why the OP is asking this question. It is likely just be a typo in the book.