1

I want to define a finite set of finite sequences and they may have distinct cardinality. Is it correct to express this as follows:

Let $S=\{x_{n_{i}}: n=0,1,...,m \quad \text{and} \quad i=0,1,...,n_i\}$

I mean for a fixed $n$ we have a sequence of $i$ elements.

Andrés E. Caicedo
  • 79,201
user113715
  • 1,101
  • 1
    @Regret: That's my reading as well; perhaps you should post your comment as an answer. (Unfortunately I didn't see your comment until I'd posted, but if you post I'll delete my answer.) – Andrew D. Hwang Mar 10 '15 at 11:49

1 Answers1

1

Normally "double subscript" notation $n_{i}$ means you have a function $n$ of an index $i$, and $n_{i} = n(i)$, which seems not to be what you require.

It sounds as if you want double indices, $$ S = \{x_{n, i}: i = 0, 1, \dots, i_{n},\ n = 0, 1, \dots, m\}. $$ In longhand: \begin{align*} &x_{0,0}, x_{0,1}, \dots, x_{0,i_{0}}; \\ &x_{1,0}, x_{1,1}, \dots, x_{1,i_{1}}; \\ &\qquad \vdots \\ &x_{m,0}, x_{m,1}, \dots, x_{m,i_{m}}. \end{align*}

Andrew D. Hwang
  • 78,195
  • Hmm, wait a minute. Wouldn't this be the set of all elements of all sequences? If the OP wishes for a set of all sequences, it would be along the lines of $$\left{(x_{n,i})_{i=0}^{i_n}~\middle|~ n=0,1,\dots,m\right}$$ – Regret Mar 10 '15 at 12:22
  • @Regret: Yes, you're perfectly correct. :) I was matching the OP's notation. (It's not entirely clear the OP is distinguishing sequences from lists of elements, but I may have mis-gauged their intent.) – Andrew D. Hwang Mar 10 '15 at 12:58