Is the disjoint union of two Matroids a Matroid itself?
Let $E_1$ and $E_2$ be two disjoint sets. Moreover, assume that $(E_1,S_1)$ and $(E_2,S_2)$ are matroids.
Define $S:=\{X \cup Y|X \subseteq S_1 \land Y \subseteq S_2\}$
Prove that $(E_1 \cup E_2,S)$ is a matroid.
I am trying to understand the solution to the problem in the post above (note that matroids $(E,S)$ in this post are defined as a ground set $E$ and a set of independent sets $S$). I know how a matroid is defined and how to check whether a given tuple of sets $(E, S)$ is a matroid. What I don't understand is how I can guarantee that $\emptyset \in S$ since there is no restriction on which subsets $X, Y$ are taken from $S_1$ and $S_2$, respectively.
As far as I understand, a subset relation $A \subseteq B$ puts no restriction on which elements of $B$ are in $A$ and $A = \emptyset$ also fulfills $A \subseteq B$. Thus, I don't know how to guarantee that $\emptyset \in S$, which is a requirement for $(E,S)$ to be a matroid according to the definition here. Therefore, to my knowledge, $(E,S)$ is not a matroid, contrary to the answer to the post above.
I would be really thankful if you could help me to understand where my reasoning is wrong.