Using mathematical symbols to express conditions

Question

There are 5 sets A,B,C,D and E.

A,B,C,D are all different from eachother.

E contains elements of at most two sets of A,B,C,D.

How can I write that with math symbols without listing all possible cases?

Normal language is allowed but should be minimized.

Example what E could look like:

$E=\{a_m,...,a_n,b_j,...,b_k\}$

Or $E=\{\}$

Answer 1

What I think the condition you're describing means is

$$ \Bigl| \{X\in\{A,B,C,D\}\mid X\cap E\ne \varnothing\}\Bigr| \le 2 $$

But this would require your example $E$ to be $\{a_m,...,a_n,b_j,...,b_k\}$, rather than $\{\{a_m,...,a_n\}\{b_j,...,b_k\}\}$.

Answer 2

$(E \subseteq \{A, B, C, D\}) \land (|E| \le 2) $

Translation:

E is a subset of {A, B, C, D}, so that every element of E is one of A, B, C, D; and ($\land$) the number of elements of E (written "|E|") is at most two.