For a recent question: "We take $A$ as a ring and $S$ a multiplicative set (not sure if this might help, but $S=\{1,f,f^2,...\}$ is the multiplicative set). Show that $\mathrm{Spec}(S^{-1}A) \subseteq \mathrm{Spec}(A)$ is the complement in $\mathrm{Spec}(A)$ of $V((f))$."
I wrote the full question in case any of the details would help, but I'm really just struggling with the line $\mathrm{Spec}(S^{-1}A) \subseteq \mathrm{Spec}(A)$, this makes no sense to me. Of course, there is a one to one correspondence between the two, under the canonical $S$-inverting homomorphism $k:A \to S^{-1}A$, but I can't really parse out why one would write that it is a subset of $\mathrm{Spec}(A)$.
It's just an abuse of notation. When you have a canonical injection from a set $X$ to a set $Y$, it is common to identify $X$ with its image in $Y$ and write $X\subseteq Y$. You are surely used to this in many other contexts. For instance, if $R$ is a ring, then $R$ is not actually a subset of the polynomial ring $R[x]$ (at least for the some common set-theoretic definitions of $R[x]$), but it is common to abuse notation and say $R\subseteq R[x]$ is the subring of constant polynomials.
I would agree that this abuse of notation is an unwise choice when posing an exercise like this that involves understanding the basic properties of this canonical injection.