$$R=\{(A,B)\mid A \subseteq B\}$$
$R$ is assumed to be a relation on a collection of sets
Since $A$ is a subset of itself, the relation is reflexive.
And if $A$ is a subset of $B$ which is, in turn, a subset of $C$, then $A$ must be a subset of $C$ making the relation transitive.
However, $A$ might not equal $B$ and in that case, the relation will not be symmetric nor anti-symmetric (since there might be some case where $A = B$ and $B$ is, therefore, a subset of $A$). I'm a little confused about this last bit. Is my thought process right or will the relation be symmetric too?
Edit: $R$ is assumed to be a relation on a collection of sets.
As you noted, it is Reflexive because: $\forall A~ (A\subseteq A)$ by definition of subset.
Likewise it is Transitive because: $\forall A~\forall B~\forall C~(A\subseteq B\wedge B\subseteq C\to A\subseteq C)$ by definition of subset.
However, it is actually Antisymmetric because $\forall A~\forall B~((A\subseteq B\wedge B\subseteq A)\to A=B)$ by definition of subset.
Next, it is only not symmetric when: $\exists A~\exists B~(A\subseteq B\wedge B\not\subseteq A)$ . Therefore, this is dependent on what collection of sets the relationship is held over, as it will only be not-symmetric when at least one set is a proper subset of another.