# Set builder notation: defining the number of elements

I have a set L and I have a subset S which is part of L and contains three elements A, B and C. Finally, each of these elements are subsets that consist of their own elements:

$A=\{a_1...a_n\}$

$B=\{b_1...b_n\}$

$C=\{c_1...c_n\}$

A, B and C contain the same number of elements and those elements stand in a binary relation.

What I want to do is to specify that A, B and C consists of 'several' elements. 'Several' here means that there is at least one element and at most the number of elements which are smaller than L.

This is my attempt:

$S=\{x_ i | 1\le\ i\ \lt\ L\}$

where I use x to denote all elements from A, B and C and i to denote the number of elements.

I think that a link is missing here between the final representation and subsets A, B and C. Any suggestion how to deal with this? Thanks.

The number of elements of a set A, called the cardinality of A is denoted $|A|$. Here's what I understand you are trying to say:
I have a set $L$ and a set $S$ which is a subset of $L$. $S$ is the set $\{A, B, C\}$, where: $$A = \{a_1, a_2, ... a_n\}$$ $$B = \{b_1, b_2, ... b_n\}$$ $$C = \{c_1, c_2, ... c_n\}$$ for some integer $n$ such that $1 \le n \lt |L|$.