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I would like to simply say "$a, b, c, d$ are all different" by mathematical notation. If I want to write all equal, then I can write $a = b = c = d$. However, if I write $a \neq b \neq c \neq d$, that does not cover $a \neq c$ and $b \neq d$ etc.

Is there a neat way to write this, or shall I go with good old $a \neq b\ \wedge a \neq c\ \wedge a \neq d\ \wedge b \neq c \dots$?

padawan
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3 Answers3

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There are a number of ways to do this, but I think the most legible one is just to say in (e.g.) English, "$a,b,c,d$ are distinct". If that doesn't work, here are some options:

  • $a \neq b \land a \neq c \land a \neq d \land b \neq c \land b \neq d \land c \neq d$
  • $|\{a,b,c,d\}| = 4$.
  • If you have an ordering available (such as when $a,b,c,d$ are numbers), and $a,b,c,d$ are arbitrary, you might say $a < b < c < d$.

Edit: I would point out that if you are being very formal, $a = b = c = d$ also doesn't work. If $a,b$ are elements for which an equality comparison makes sense, then $a = b$ is a proposition -- you cannot apply any further relational symbols to a proposition. The reason we "allow" $a = b = c = d$ is that everyone understands what you mean anyway.

Mees de Vries
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  • Thank you for clarification. Then it is best to write "if $a,b,c,d$ are all equal or all different," I suppose. – padawan Apr 18 '17 at 17:50
  • @padawan, not if $a,b,c,d$ are arbitrary. It is possible that $a = b$, while $a,c,d$ are distinct. If you want to make no assumptions on equalities holding between $a,b,c,d$, you don't need to comment on it at all, or if you want to emphasize that they could possibly be equal, you could say "let $a,b,c,d$ be [somethings], not necessarily distinct". – Mees de Vries Apr 18 '17 at 17:52
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Often you see this done, if the variables are subscripted, as: $x_i\ne x_j$ for $i\ne j$. But that would require changing your notation, which, I realize, may not be desirable.

paw88789
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We can use the phrase $a,b,c,d$ are pairwise different to indicate that no two elements are equal.

See e.g. this related MO post.

Markus Scheuer
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