Mathematical notation for distinct numbers

Question

How can I represent a set of distinct numbers in shorthand mathematical notation? In particular I am trying to write that there are 4 distinct integers $a,b,c,d$.

For three numbers, I could say:

$a,b,c \in \mathbb{Z} :a \not= b\not= c \not=a$; however, for larger sets, I need to compare every pair of numbers for a total of $\frac{n!}{2}$comparisons.

Is there an accepted notation for distinctness? Or could I make a statement such as $a,b,c,d \in \mathbb{Z}, \{a,b,c,d\}$ is a set, bearing in mind that sets cannot have duplicates?

Answer 1

Hint: Recall that both representations \begin{align*} \{1,1,2,3\}=\{1,2,3\} \end{align*} are valid and denote the same set. So this is not the appropriate notation for your needs.

It's common to use the phrase pairwise distinct for instance

• Let $a,b,c,d\in\mathbb{Z}$ denote pairwise distinct integers which ...

The idea behind this phrase is that whenever we select two of the elements under consideration then they are different.