On the definition of the Minkowski sum

Question

Closed sets need not be mapped to closed sets by the Minkowski sum as the following example shows: $$ S_1 := \{ (x, y) \mid x, y \in \Bbb R, x \geq 0, x y \geq 1 \},$$ $$ S_2 := {\Bbb R} \times \{0\} $$ and $$ S_1 + S_2 = \{ (x,y) \mid x \in {\Bbb R}, y > 0 \} $$

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Here on the above example, I don't quite understand the definition of such a set, and the consequence of the Minkowski sum. Could you please help me to figure this out?

Answer 1

The definition of sum of two set is usually taken to be

$$S_1+S_2=\{s_1+s_2\mid s_1\in S_1, s_2\in S_2\}$$

Which is the set of all possible sums where you take one element from each set and add them.

To see why they are the same. Take any $(x,y)$ where $y>0$, rewrite it as $(\frac 1y + (x-\frac 1y), y)=(\frac 1y,y) + (x-\frac 1y,0)$, where the first point is in $S_1$ and second point is in $S_2$. This shows that $\{(x,y)\mid x\in \mathbb R, y>0\} \subset S_1+S_2$. Similarly you can prove the other inclusion, thus showing the two sets are equivalent.

Answer 2

The product of sets is

$$ A \times B = \{ (a,b) \mid a \in A, b \in B \} $$

therefore

$$ \mathbb{R} \times \{ 0 \} = \{ (a,0) \mid a \in \mathbb{R} \} $$