How to write a subset with index

Question

I have a difficulty to define the start and end index of a subset.

I have $x_i$, $i=1, ..., |\phi_M|$ an element of the main set $\phi_M$, Now I have to define subsets of $\phi_M$ but I have doubt on what is the correct form,

$A=\{x_j, j=1,...,|\phi_M|\}$

$A=\{x_j, j=l,...,k, 1 < l< k<|\phi_M|\}$

$A=\{x_j, x_j \in B, B \subset \phi_M \}$

What is the correct form? or how can I correct please?

Answer 1

The way I've seen it done is to specify $\{i_1, ....., i_k\}\subset \{1,...., |\phi_M|\}$ and $A = \{x_{i_j}|x_{i_j} \in \phi_M\}$.

In fact just $A=\{x_{i_j}\}\subset \phi_M$ is usually understood that the $j$ is is the index for $A$ and runs $1,.... , |A|$ and that $i_j $ is an index element from the indexes of $\phi_M$.

Example: $\phi_M = \{x_1,....., x_m\}=\{x_i\}$ and $A = \{x_2,x_3, x_5, x_{13}\} = \{x_{i_1},x_{i_2},x_{i_3},x_{i_4}\}=\{x_{i_j}\}$ where $i_1 = 2; i_2 =3; i_3 = 5, i_4=13$.

Answer 2

Define $I = \{1,\dots,|\phi_M|\}$ and put $$A = \{\{x_i\}:\, i\in I\} \cup \{\{x_i,x_j\}:\, (i,j)\in I^2\} \cup \cdots \cup \{\{\{x_{i_1}\},\dots,\{x_{i_k}\}\} :\, (i_1,\dots,i_k)\in I^k, \, k=|\phi_M|-1\} \cup \phi_M$$