How to find proper subsets $S_1, S_2, S_3$ (that are different from $\{(0,0)\}$ and $\varnothing$) of vector space $\mathbb{R}^2$ so that $S_1+S_1\subsetneq S_1,\\ S_2\subsetneq S_2+S_2,\\ S_3+S_3=S_3?$
How to find proper subsets of vector space $\mathbb{R}^2$ that fulfill the following requirements?
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linear-algebra
vector-spaces
1 Answers
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For $S_1$: e.g. $S_1 = \{(x,y) \in \mathbb{R}^2 \ | \ x \in \mathbb{N}, y=0 \}$ (here $(1,0) \in S_1$ but $(1,0) \not \in S_1 + S_1$)
For $S_2$: e.g. the unit ball $B_1(0)=\{x \in \mathbb{R}^2 \ | \ |x| \leq 1 \}$
For $S_3$: e.g. a $1$-dimensional linear subspace
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0Yes. Consider the two vectors $a=(\frac{1}{2},\frac{\sqrt{3}}{2})$ and $b=(\frac{1}{2},-\frac{\sqrt{3}}{2})$. They have length 1 and when you add them, you get the vector $(1,0)$. Obviously you can reach then every other point on the unit circle by rotating the vectors $a$ and $b$. Hence $S_2 \subset S_2 + S_2$. That $S_2 \not = S_2 + S_2$ is trivial. – 2017-02-10