I have a sequence of sets where each set is given as $\mathcal{M} = \{m_i \mid 1 \leq i \leq n\}$. What would be the correct notation for the sequence of sets?
- Is $\mathcal{S} = \langle \mathcal{M}_j \mid j \in \mathbb{N} \rangle$ correct?
Then out of $\mathcal{M}_j = \{m_{j,i} \mid 1 \leq i \leq n\}$ or $\mathcal{M}_j = \{m^j_i \mid 1 \leq i \leq n\}$, which one would be appropriate?
According to set theory, an ordinal $j$ is the set of all smaller ordinals, i.e.
$$j = \{i : i < j\} = \{0, 1, 2, \dots, j-1\}$$
Let
$$\mathcal{S} = j \mapsto m[j]$$
where $f[A]$ denotes the image of $A$ under $f$. Then
\begin{align} \mathcal{S}(j) &= m[j] \\ &= \{m(i) : i \in j\} \\ &= \{m(i) : i < j\} \end{align}
If you start indexing at 0 rather than 1 (which is convenient for reasons like the above), this seems to be what you want.