I would appreciate it if anybody shows me the most compact mathematical notation for the following sets formed by operations, so the subscript is different in every "e"($e_A$, $e_B$, $e_C$ are different sets).
- ${e_{_{i}}}\cap\ {e_{_{i}}}\cap\ {e_{_{i}}} \ |_{i\in \{A,B,C\}}$
- ${e_{i}}\cap\ {e_{i}}\cap \ {{e^c}_{i}} \ |_{i\in \{A,B,C\}}$
- ${e_{i}}\cap \ {{e^c}_{i}} \cap \ {{e^c}_{i}} \ |_{i\in \{A,B,C\}}$
Ex: I've tried with ${e_{i}}\cap \ {{e^c}_{j}} \cap \ {{e^c}_{k}}\ |_{i,j,k\in \{A,B,C\}| i\neq j\neq k }$ for the last one, but I don't know if this is correct and if it's the most compact expression.
PD: I'm not asking about giving a number to each subscript but express without ambiguities the example above.
EDIT: What I want to say is, there are the following sets, $e_A$,$e_B$,$e_C$ and I need to say, for example, $e_A \cap e_B \cap {e^c}_C$, $e_B \cap e_C \cap {e^c}_A$, $e_C \cap e_A \cap {e^c}_B$. Then how can I write this? ${e_{i}}\cap\ {e_{j}}\cap \ {{e^c}_{k}} \ |_{i,j,k\in \{A,B,C\}\ \wedge\ i\neq j \wedge j\neq k \wedge i\neq k}$ ?
Maybe it's a really simple question, but thanks in advance.