Dissimilar Terms

Probability Level 4

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

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

Note that "ab,ac,bca \neq b, a \neq c, b \neq c" is not valid because it is three statements, not one, and that "(ab)(ac)(bc)0(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,da,b,c,d are all distinct numbers" in such a way?

Bonus: Generalize this.


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