# Shorthand set builder notation?

$$\big\{k\big\}_{k=0}^n$$

I assume this implies that

$$\big\{k\big\}_{k=0}^n = \{k \in \mathbb{WHAT} : 0 \leq k \leq n\}$$

What is $$k$$ an element of? I assume integers, but it might as well be real or complex. I propose a better, less ambiguous notation:

$$\big\{k \in \mathbb{SOMETHING}\big\}_{k=0}^n$$

Would this work better or does this mean something completely different?

• It might also refer to a sequence. Commented May 22, 2022 at 13:41
• Could you tell us which papers you found this notation in? Commented May 22, 2022 at 13:51
• – VJZ
Commented May 22, 2022 at 13:56
• IMO it is simply $\{ 0,1,\ldots , n \}$ Commented May 23, 2022 at 6:52

The notation $$\{f(k)\}_{k=0}^n$$ in the context given means either the sequence or set $$f(0),f(1),f(2),\dots,f(n)$$
in analogy to sum or product notation, where $$\sum_{k=0}^nf(k)=f(0)+f(1)+\dots+f(n)$$
and $$\prod_{k=0}^nf(k)=f(0)\times f(1)\times\cdots\times f(n)$$
Your notation $$\{k\}_{k=0}^n$$ would mean $$\{0,1,\dots, n\}$$, but I think the paper is actually using examples like $$\{\mathbf{c}_j^{(t)}\}_{j=1}^k=\{\mathbf{c}_1^{(t)},\mathbf{c}_2^{(t)},\dots,\mathbf{c}_k^{(t)}\}$$.
Instead of your notation, what you might see instead is $$\{f(k)\}_{k\in S}$$ for some set $$S$$, again in analogy to sum or product notation.