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I have a pretty dumb question. I know that if I have a pair $(a,b)$ that can take values in $\mathbb{R}$, I should write $(a,b) \in \mathbb{R}^2$. But what about if I have a set? I want $\{a,b,c\}$ to all take values in $[1,3] = \{1,2,3\}$. Do I write $\{a,b,c\} \in [1,3]^3$?

Somehow it seems odd to me.

Thanks

excalibur1491
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2 Answers2

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In the example you mentioned, if you have an (unordered) set, e.g. $\{a,b,c\}$, then it is customary to write $$ \{a,b,c\} \subset [1,3]. $$ The notation $\subset$ means "is a subset of" as in $$ \mbox{ a smaller set } \subset \mbox{ some larger set} $$ On the other hand, if you have an ordered pair or ordered triple (and so on), then it is customary to write the pair elements $(a,b)$ in parentheses and use the $\in$ and $\times$ symbols, as in $$ (a,b) \in {\mathbb R}\times{\mathbb R}. $$

Alex
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  • That is what I have seen elsewhere, but it's .. dodgy, isn't it? It means that ${a,b,c}$ is a subset of ${1,2,3}$. I guess, ${3,1,2}$ is also a subset, so it induces all the possible values of a,b, and c. I get it, but it just seems odd. But thanks, that is what I had seen, and you confirmed it. (also, could be $\subseteq$). – excalibur1491 Apr 17 '17 at 01:59
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Yes that's right. People will also write $(a,b) \in A \times A$ or $(a,b,c) \in A \times A\times A$. You can also write something like "consider the pair $(a,b)$ where $a, b \in A$" (the last bit is read as "where $a$ and $b$ are in $A$).

Trevor Gunn
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