Is there any mathematical notation to expand a list/set of values?
I am looking for something that would expand like this:
$A=(x=0,x<=3)[2x+1]$
$A=[2(0)+1,2(1)+1,2(2)+1,2(3)+1]$
$A=[1,3,5,7]$
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Maybe $$ \big[2n+1\big]_{n=0}^3 $$ – GEdgar Oct 08 '22 at 16:10
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A compact notation for the corresponding set would be $2(\Bbb Z\cap [0,3])+ 1$ or $(2\Bbb Z+1)\cap [0,7]$. – LL 3.14 Oct 08 '22 at 23:22
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If it is an ordered list, then it is denoted with parentheses: $(2n+1)_{n\in{0,1,2,3}}$ – LL 3.14 Oct 08 '22 at 23:24
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Yes, the counterpart of sigma or pi notation for sequences is typically notated as $\left(a_n\right)_{n\in S}$ where $a_n$ expresses the $n$'th term and $S$ is an index set. For general sets, a different notation is usually used, called set builder notation, which has the form $\left\{x|\phi(x)\right\}$, where $\phi$ is some formula satisfied by the included elements $x$. Depending on the system we are using, there may be some limitations on the sets that are allowed to be built, hence which expressions in this notation are valid.
So, the even numbers, $0,2,4,6,\ldots$, could be expressed in the above notations as an (indexed) sequence as $\left(2n\right)_{n\in\mathbb{N}}$ or as a (non-indexed) set $\left\{x|\exists y \in \mathbb{N} . x=2y \right\}$.
Jam
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We use finite sequences to denote such lists: \begin{align*} A=\left(2x+1\right)_{0\leq x\leq 3}=(1,3,5,7) \end{align*}
Markus Scheuer
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@LL3.14: It is a common notation. Here $x$ is a bound index variable as well as any other like $n$ or $q$ or $t$. The slight advantage here is to have an expression $2x+1$ which is also stated in OPs question, just to enhance readability. – Markus Scheuer Oct 08 '22 at 23:34
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Yes, but the same notation is also common for real numbers ... $(2x+1)_{x\in{0,1,2,3}}$ would be more standard I think. – LL 3.14 Oct 08 '22 at 23:47
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@LL3.14: I can partly agree. The drawback with the set notation is we loose the order property which is also crucial for $n$-tuples. Nevertheless, both notations are commonly in use, since we can often read for example $(a_n){n\geq 1}$ as well as $(a_n){n\in \mathbb{N}}$. – Markus Scheuer Oct 09 '22 at 09:30