TITLE: Interpretation of "not equal" notation
QUESTION [8 upvotes]: This will be a short question. Let $x$, $y$, $z$ be three elements from any set. Is the following:
$$x \ne y \ne z \tag{1}$$
Equivalent to:
$$x \ne y, ~ ~ y \ne z, ~ ~ z \ne x \tag{2}$$
Or simply:
$$x \ne y, ~ ~ y \ne z \tag{3}$$
Is it even well-defined? I know what $x = y = z$ implies but what about negation? I know I could use $x \ne y \ne z \ne x$ to "make sure" but I was interested in knowing what $(1)$ really means.

REPLY [7 votes]: We do write often e.g. $x \in y \in z$ as a familiar and acceptable shorthand for $x \in y \land y \in z$. That's evidence that we do allow this kind of notational collapse for conjoined relational propositions even when the relation isn't transitive. (We don't read $x \in y \in z$ as implying $x \in z$.)
Can we similarly write $x \ne y \ne z$ as shorthand for $x \ne y \land y \ne z$, again with no implication that $x \ne z$? You might predict, by parity, that we would: yet I certainly don't think this is common practice. Why not? What makes the difference? Dunno!
I think I have, however, seen the likes of  $x \ne y \ne z \ne w $ used as local shorthand for the corresponding symmetric $x \ne y \land x \ne z \land x \ne w \land y \ne z \land y \ne w \land z \ne w$. But I wouldn't adopt such a shorthand without saying what you are doing!