Sum of sets of vectors (or whatever it's called)

S+T = {s+t | s $\displaystyle \in$ S, t $\displaystyle \in$ T}

Can someone please explain this property better?

Let's say S = {(1,0),(0,1)} and T = {(2,1),(1,3)}

What would be S+T?

I would just assume that S+T would be each vector from S plus each vector from T:

S+T = {(3,1),(2,3),(2,2),(1,4)}

But obviously I'm wrong since S+T is a subspace and what I have is clearly not a subspace. How do you get the 0 vector from S+T?

Thanksss.

***Correction - S+T is a subspace only if S and T are both subspaces, and that would explain the zero vector thing... So is my calculation correct?? And does that mean that in this particular example S+T is not a subspace??