Dissimilar Terms

We can rewrite the statement "\(a\) and \(b\) are distinct numbers" into a statement made of just variables and not-equal signs: "\(a \ne b\)".

Similarly, we can rewrite the statement "\(a,b,\) and \(c\) are distinct numbers" as "\(a \ne b \ne c \ne a \)".

Note that "\(a \neq b, a \neq c, b \neq c\)" is not valid because it is three statements, not one, and that "\((a-b)(a-c)(b-c) \neq 0\)" is not valid either because it has something other than variables and not-equal signs, which in this case are the parentheses.

In the above two statements, the minimum numbers of not-equal signs \((\ne)\) used are 1 and 3, respectively.

What is the minimum number of not-equal signs used to rewrite the statement "\(a,b,c,d\) are all distinct numbers" in such a way?

Bonus: Generalize this.

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