Set $S=\{(x,y,z)| x,y,z \in \mathbb{Z}\}$ is a subset of vector space $\mathbb{R}^3$, how do I show that it is not a subspace of $\mathbb{R}^3$. So I know that set  $S=\{(x,y,z)| x,y,z\in \mathbb{Z}\}$ is a subset of vector space $\mathbb{R}^3$.
Specifically, it is worded in our lecture that it is a " subset of $(\mathbb{R}^3, \oplus, \odot)$ , where $\oplus$ and $\odot$ are the usual vector addition and scalar multiplication."
My teacher has stated in our lecture that this set $S$ is not a subspace of $\mathbb{R}^3$.
But from what I can tell $S$ is:


*

*Closed under addition

*Closed under multiplication

*Contains a zero vector $(0,0,0)$


How is it not a subspace of $\mathbb{R}^3$, what am I missing?

Edit: User Randall pointed out that I misread the question. (I assumed S was under some invalid field.)
First lets look at the definition of a subspace:


*

*All products and sums composed of elements within the subspace also are in the subspace.

*All elements in the subspace must be able to be scaled by the vector space surrounding the subspace.

*0 is also an element of the subspace.
You remembered rule 1 and 3 however, it's clear S is violating rule 2.
In order for S to be a subset of $R^3$, all elements within S that are scaled by any number within R are also an element of S.
