So lets consider the following grammar
$$ \begin{align*} S &\to 0 \mid 0A \\ A &\to 1 \end{align*} $$
would the string "1" be accepted by the language or must the language start with $S$?
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Sign up to join this communitySo lets consider the following grammar
$$ \begin{align*} S &\to 0 \mid 0A \\ A &\to 1 \end{align*} $$
would the string "1" be accepted by the language or must the language start with $S$?
No $1$ will not be a part of the grammar, as only those strings generated by starting from the start variable $S$ are part of the grammar.
What you have shown is technically not a grammar, only part of it. A grammar is formally defined as the tuple $(N, \Sigma, P, S)$, where:
You have only provided $P$, but to have a grammar, you also need $N$, $\Sigma$ and $S$.
$N$ and $\Sigma$ are often omitted when defining a grammar, because they are clear from $P$ (like in your example, where $N = \{ S, A \}$ and $\Sigma = \{ 0, 1 \}$).
Explicitly specifying the start symbol (referred to as $S$ in the formal definition above) can be omitted when there is a clear convention for the name of the start symbol; naming it $S$ as you do in your example is a common convention.
What this all means for your example is that if you assume that $S$ is the start symbol, then $1$ is not a member of the language. If you instead assumed that $A$ is the start symbol, then $1$ would be a member of the language. Such grammar would be formally valid, but defining it like this wouldn't make sense from a human's point of view.