Does $a≠b≠c$ imply $c≠a$? Is $a≠b≠c$ a shorthand for ($a\neq b$ and $b\neq c$), or is it a shorthand for ($a\neq b$ and $b\neq c$ and $c\neq a$)?
I want to know the "correct and formal" answer, rather than the "most commonly used" answer.
 A: Summarizing comments:
$\neq$ is not a transitive relation.  As such, $a\neq b$ and $b\neq c$ does not directly imply that $a\neq c$.
There are some who interpret $a\neq b\neq c$ to imply all of $a,b,c$ are pairwise distinct.  There are others who interpret $a\neq b\neq c$ to instead imply only the two statements $a\neq b$ and $b\neq c$ which does not imply anything about the relationship between $a$ and $c$.
In the end, the notation itself is generally considered informal and should be avoided and if you see it used you should infer from context which is meant or ask for clarification.

Suggestions for writing that variables $a_1,a_2,a_3,\dots,a_n$ are all distinct:

*

*Order them: "Let $a_1,a_2,\cdots, a_n$ be real numbers such that $a_1<a_2<a_3<\cdots a_n$"  (only works in an ordered field)

*Use words: "Let $a_1,a_2,\cdots, a_n$ be distinct real numbers"

*Use quantifiers: "Let $a_1,a_2,\cdots, a_n$ be real numbers such that $\forall i\neq j$ we have $a_i\neq a_j$"

*Write each: "Let $a_1,a_2,a_3$ be real numbers such that $a_1\neq a_2,~a_1\neq a_3$ and $a_2\neq a_3$" (impractical with more variables)

