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- Apr 14, 2013

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I want to find subsets $S$ of $\mathbb{R}^2$ such that $S$ satisfies all but one axioms of subspaces.

A subset that doesn't satisfy the first axiom: We have to find a subset that doesn't contain the zero vector. Is this for example $\left \{\begin{pmatrix}x \\ y\end{pmatrix} : x,y>0\right \}$ ?

A subset that doesn't satisfy the second axiom: We have to find a subset that doesn't contain the sum of two vectors of $S$. Could you give me an example for that?

A subset that doesn't satisfy the third axiom: We have to find a subset that doesn't contain the scalar product of a vector of $S$. Is this for example $\left \{\begin{pmatrix}x \\ y\end{pmatrix} : x\geq 0\right \}$ since $\begin{pmatrix}1 \\ 0\end{pmatrix}\in S$ but $(-1)\cdot \begin{pmatrix}1 \\ 0\end{pmatrix}\notin S$.