More on the Definition of a Set That Is Shattered by a Class of Sets

This post reviews the definition a set that is shattered by a class of sets and discusses several different ways to look at the definition. Let's take the following def. of a shattered set from Wikipedia:

Suppose $A$ is a set and $\mathcal{C}$ is class of sets. $A$ is shattered by $\mathcal{C}$ if any subset of $A$ is equal to the intersection of some element of $\mathcal{C}$ and $A$. That is:

$$ A\text{ is shattered by }\mathcal{C}\leftrightarrow(\forall a\in\mathcal{P}(A))(\exists c\in \mathcal{C})(a=c\cap A), $$ where $\mathcal{P}$ denotes the power set of $A$.

From the definition, it's easy to sea that if $A$ is shattered by $\mathcal{C}$ then:

$\forall a\in\mathcal{P}(A)\rightarrow (\exists c\in\mathcal{C})(a\subseteq c)$

However, the definition requires more than just $(\exists c\in\mathcal{C})(a\subseteq c)$ but rather $(\exists c\in \mathcal{C})(a=c\cap A)$. That is, not only $c$ contains all elements in $a$ but it may have extra elements not in $a$, however, those extra elements can't be in $A$. And this is a very important thing to remember.

If we define the class $\mathcal{C}$ restricted to $A$, denoted by $\mathcal{C}_{A}$, is the class that contains all set $c\cap A$ for all $c\in\mathcal{C}$. That is: $\mathcal{C}_{A}=\{c\cap A: \forall c\in\mathcal{C}\}$, then another way to phrase $A$ is shattered by $\mathcal{C}$ is as follows:

$\forall a\in \mathcal{P}(A)\rightarrow(\exists c_{A} \in \mathcal{C}_{A})(a=c_{A})$.

From the definition of $\mathcal{C}_{A}$, we have that if $c_{A}\in\mathcal{C}_{A}$ then $c_{A}$ is a subset of $A$. So a third way to say that $\mathcal{C}$ shatters $A$ is if $\mathcal{C}_{A}=\mathcal{P}(A)$.