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The sum of two sets of numbers $S,T$ is defined as: $$ S+T=\big\{z \mid \exists x \in S \quad\exists y \in T \text{ such that } x+y = z\big\} $$

I'm not really sure what $S+T$ is. If $S=\{s_1,s_2,\ldots,s_n\}$ and $T = \{ t_1, t_2, \ldots,t_n\}$ then is $S+T=\{s_1+t_1,s_2+t_2,\ldots,s_n+t_n\}$? Is $z=S+T$ or $S+T=\{z_1,\ldots,z_n\}$?

Michael Hardy
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Yos
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    The definition says it is all possible sums $x+y$ with $x \in S$ and $y \in T$. It does not say that your subscripts have to match for some reason (nor does it say subscripts even have to be there). – Randall Jan 29 '18 at 18:13
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    @Randall if $S={2,3}$ and $T={4,5}$ then $S+T={2+4, 2+5, 3+4,3+5}$? – Yos Jan 29 '18 at 18:15
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    With set braces and without redundant $7$s, yes: ${6, 7, 8}$ – Randall Jan 29 '18 at 18:15

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The notation is obfuscating. Another way to write the same thing is

$$S+T = \{x+y\;|\;x\in S, y\in T\}$$

It's just the set of all possible sums resulting from adding an element of $S$ to an element of $T$.

Sometimes it's easier to see visually that it's the same as $$\bigcup_{s\in S}(s+T)$$

The sets $s+T$ are copies of $T$ that have been shifted by $s$. Then just combine all of these shifted copies.

I'm assuming here you are working with subsets of $\mathbb R^n$.

MPW
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  • What is the small $s$ and "shifted copies"? – Yos Jan 29 '18 at 18:23
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    @Yos The lower case $s$ is an element of $S$, and $\textit{shifted copies}$ means the cardinality of $s+T$ equals the cardinality of $s'+T$, for any $s,s'\in S$: the sets $s+T$ and $s'+T$ are not equal but there is a bijection between them. – Mee Seong Im Jan 29 '18 at 18:28