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Is $(a, b, c, d) \in\mathbb{R}$ an adequate way to say to "a, b, c, d is a list of real numbers?"

Or would it be better to say: Given list $(a,b,c,d)$ where $a,b,c,d \in\mathbb{R}$ ?

I am pretty clear on set notation but my search on list notation has been inconclusive.

jmxdbx
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2 Answers2

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The latter is standard when the numbers are being considered separately. The former is probably what you want, but you need to be careful. $\mathbb{R}$ is a set, but it doesn't contain any lists - it only contains individual numbers as its elements.

When you write a list of numbers or symbols in round parentheses, most mathematicians would call that an $n$-tuple. Typically we would write $$(a, b, c, d)\in \mathbb{R}^4$$ Here $\mathbb{R}^4$ means the product of four copies of the real numbers; this is the set of all possible $4$-tuples, where each entry is a real number.

Summary: close, but no cigar.

Alexander Heyes
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  • Thanks. Yeah I realize that "subset" and "element of" are different concepts, I guess I was looking for some sort of sublist of a set symbol, but I'm just using the latter option of my initial post for now. What if it's positive integers? So $(a_0, a_1, a_2, \ldots, a_n) \in\mathbb{Z}^n$ would make sense? but $(a_0, a_1, a_2, \ldots, a_n) \in\mathbb{Z}^{+n}$ seems unclear. – jmxdbx Jan 17 '16 at 23:09
  • @jmxdbx if you're starting from 0, it's $(\mathbb Z^+)^{n+1}$. Zero based indexing is a good way to make mistakes and really only exists because it was used as an offset to address memory. – Matt Samuel Jan 17 '16 at 23:15
  • @MattSamuel Yes good point, my mistake on the $n + 1$. And thanks, I was thinking use of parenthesis might clarify. – jmxdbx Jan 17 '16 at 23:17
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In model theory, it is common to write $\overline{x}$ for a list of elements $\langle x_1, \ldots, x_n\rangle$ (or $(x_1, \ldots, x_n)$, if you prefer) and then to write $\overline{x} \in X$ to mean that each $x_i \in X$. Apart from that abuse of notation in model theory, $(a, b, c, d) \in \Bbb{R}$ only makes sense if you have somehow defined $\Bbb{R}$ to be a set of 4-tuples and are talking about the internal details of those 4-tuples (which is almost certainly not the case in normal mathematical writing).

Rob Arthan
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