Notation for multiple elements not equal to each other

Question

I have four elements, $A,B,C$ and $D$. I want to write that none of the elements are equal to each other. Basically, something like this:

$$A \neq B \neq C \neq D$$

However, this is incorrect, since $\neq$ is not transitive.

What would be an elegant and correct way to write this?

Answer 1

There is no elegant symbolic way to do this with your set. If you instead have an indexed set $\{a_1, a_2, \ldots, a_n\}$, you could say

$$ a_i\neq a_j \text{ when } i\neq j $$

or something to that effect.

But in your case I would just say

I have four distinct elements $A, B, C, D$.

Maybe even go so far as saying "pairwise distinct".

Answer 2

Here's one way to do it if the number of items is small enough that you can count them. You can organize the items into a set and then insist that the cardinality of that set is the same as the number of symbols.

$$ |\{ A, B, C, D \}| = 4 $$