Comparing the closed behaviors corresponding to finite state automata

Question

Let $G$ be a finite state automaton (FSA) with transfer function $\delta(\cdot,\cdot)$ and initial state $q_0$. Suppose also $\Sigma_{G}$ represents its alphabet. Assume that its closed behavior is a set of strings defined as:

$L(G) := \{s \in \Sigma_{G}|\delta(q_0,s) \text{ is defined in }G\}$

Now, consider two particular FSA $G_1$ and $G_2$ such that $\Sigma_{G_{1}} \subseteq \Sigma_{G_{2}}$. Clearly, then we clearly have $\Sigma_{G_{1}}^{*} \subseteq \Sigma_{G_{2}}^{*}$. I'm wondering whether or not we necessarily conclude that $L(G_{1}) \subseteq L(G_{2})$.

To me, it's not always true. Am I right?

Here is a simple example to contradict it:

in which $\Sigma_{G_{1}}^{*} \subseteq \Sigma_{G_{2}}^{*}$, but $L(G_{1}) \supseteq L(G_{2})$.

Answer 1

Your claim implies that $L(G)$ depends only on the alphabet of $G$, but this is clearly false. So the claim itself must be false.