What does $a \ne b \ne c$ mean?

Question

Does

1. $a\ne b \ne c$

mean

2. $a \ne b \land b \ne c$

or

3. $a \ne b \land b \ne c \land a\ne c $ ?

These are two distinct statements when 2 $\;\not\!\!\!\implies(a \ne c)$, yet I am unaware if one is "canon" usage. I have thus far always interpreted it as 3, but perhaps some of the times people have meant it as 2 without me being aware.

Answer 1

Unless the Intention is to confuse , do not write $ a \ne b \ne c $.
It means $ ( a \ne b ) \land ( b \ne c ) $ , nothing more.
It is true that $ 1 \ne 2 \ne 1 $.
The Issue is that $\ne$ is not transitive.

It is ok to write that with other transitive relations :

• $ a = b = c $ [automatically $a = c$]
• $ a < b < c $ [automatically $a < c$]
• $ a > b > c $ [automatically $a > c$]

You could write the Original Claim in various ways to suit various contexts :

• $ a \ne b \ne c \ne a $ [[ will not work for 4 or more variables ]]
• No two of $a,b,c$ are equal
• Every Pair of $a,b,c$ are unequal
• Distinct $a,b,c$
• $\{a,b,c\}$ has 3 elements
• ETC

Here is one Intentional usage to confuse :

Puzzle Question : We are given $abc=12$ ( $a,b,c$ Positive Integers ) , with $a \ne b \ne c$
What is the largest value for $b$ ?

Paradoxical Answer : $b=12$ !
We will generally assume that $1\times6\times2=12$ will give largest $b=6$ , though it should be $1\times12\times1=12$ which will give largest $b=12$ !
That is because $(a,b,c)=(1,12,1)$ , where $1 \ne 12 \ne 1$ & $abc=12$ !