I have a set $\mathbb{A}$ and an equivalence relation $\sim$ defined on that set. Now I take a subsets $A \subset \mathbb{A}$. Can I just say $A/\sim \subset \mathbb{A}/\sim$?
As I understand it I can't, since there can be $\mathbb{A} := \{x,y\}$ with $x \sim y$ while $A := \{x\}$ ($x,y \in \mathbb{A}$, $x \in A$, $y \notin A$). From this follows $\{\{x,y\}\} = \mathbb{A}/\sim$ but $\{\{x\}\} = A/\sim$ and we know $\{\{x\}\} \not\subset \{\{x,y\}\}$.
This especially comes into play, when I want to do set operations on subsets of $\mathbb{A}$ and I want to have the intersection of two subsets in regard to the equivalence classes.
Disclaimer: I hope the title actually describes what I'm asking. I'm happy about any hint for clarification.
Strictly speaking, the equivalence relation on $A$ is not $\sim$, but is the restriction of $\sim$ to $A$, i.e. the relation $\sim_A$ defined for $x,y \in A$ by $x \sim_A y$ if and only if $x \sim y$. (But this is just me being pedantic.)
As you identify, it's certainly not the case that $A/{\sim_A} \subseteq \mathbb{A}/{\sim}$. However, it is the case that, for all $U \in A/{\sim_A}$, you have $U \subseteq \mathbb{U}$ for some $\mathbb{U} \in \mathbb{A}/{\sim}$. Maybe this is the relationship between $A/{\sim_A}$ and $\mathbb{A}/{\sim}$ that you were trying to identify?
To the best of my knowledge, there isn't a succinct symbol for this; it's an example of a subquotient.