Does "V contains S" have two different meanings?

Question

Talking in terms of sets, I would take the above to mean $S \in V$. But my course's notes says

Let $S$ be a subset of a vector space $V$, the span of $S$, denoted $Span(S)$ is the smallest subspace of $V$ that contains $S$.

Which confuses me, because $V$ contains vectors, whereas $S$ is a set, not a vector, so by my definition, $V$ cannot contains $S$.

So by "$V$ contains $S$" I assume it means $S \subseteq V$, right? Is this considered correct also?

Answer 1

Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $y\in x$ or $y\subseteq x$. Some authors make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some authors do just the opposite; e.g., quoting from p. 33 of C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–37:

To avoid confusion, we shall say that a set includes its elements and contains its subsets.

Answer 2

$S$ is a set of vectors in $V$ which is not necessarily a subspace of $V$. So you can for example have $S= \{v_1,v_2,...\}$ However, when you take all linear combinations of vectors in $S$, you get the $span(S)=\{a_1v_1+a_2v_2+...|a_i \in F,v_i \in S\}$, here $F$ is the underlying field of $v$, which is a subspace of $V$.

I mean you could have $S \subset V$. Take for example $V=\mathbb{R}^3$ and $S=\{(1,0,0),(0,0,1)\}$, $Span(S)=x-z plane$.

Answer 3

I think this is an example of a function being implicitly applied to a set of its possible inputs in a point-wise fashion. Namely, since it is clear what it means for an element s of span(S) to be contained in V, we can (by point-wise extension) also give meaning to the statement that span(S) is contained in V.

The function this is applied to in this case is simply: $$\in_V : S \to \mathrm{Bool} : s \mapsto [s \in V].$$