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Possibly a strange "why would you want to" question, but how do you mathematically express a set of values that are all different?

For example, a set of values for which a mode calculation is impossible due to each element being unique.

Perry Collier
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    "Pairwise distinct" is a typical phrase that means "no two are the same." I realize that's not a notation. I would not use a notation except in case of absolute necessity (such as programming an automated proof solver). – David K Nov 01 '22 at 00:00
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    I have seen $a\neq b\neq c$ to express that $a$, $b$ and $c$ are all different. I don't really like it, since $\neq$ is not transitive. – Taladris Nov 01 '22 at 00:15
  • Many thanks guys – Perry Collier Nov 01 '22 at 00:17
  • It's better to be clear than to be short. I just write: Let ${a_i}$ be a set of values so that $a_i \ne a_j$ unless $i =j$. Of course that assumes it's a countable set. But the term "set" by default means each value is only represented once so it'd be redundant to point it out.... I truly hate $a\ne b \ne c$ as $\ne$ is not transitive nor associative and such a string of characters. If I had a long list (more than $3$) I'd just say "suppose we have $a_1, a_2, ....,a_n$ and they are all pairwise distinct" or "and for any pair $i\ne j$ we have $a_i \ne a_j$" – fleablood Nov 01 '22 at 06:19

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A set, by definition has unique elements. For e.g., $x \in \mathbb{Z}$ means $x$ is an integer, we don't have to disambiguate further.

vvg
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  • Very very true! But... there can be a case where one has a sequence or a list where you want to express the values are distinct. .... Any answer I know (I'd simply say $a_i \ne a_j$ except when $i=j$) would assume the list or sequences are countable, but then an uncountable list or sequence is meaningless. – fleablood Nov 01 '22 at 06:23
  • Not necessarily. For eg: Let $p_k$ be the $k$-th prime. This defines an uncountable sequence. If I say $\mathbb{P} = {p_k : k \in [0,\infty]}$, the elements are unambiguous. In situations like what you describe, we can just define the sequence, say $x_n$ separately and $S$ separately using set builder notation and then say $x_n \in S$ to convey the domain. – vvg Nov 01 '22 at 06:39
  • I'm not following. What is $p_\pi$, the $\pi$th prime? Certainly a sequence of all primes is countable. It is a subset of the natural numbers after all. I'm not certain what you are trying to say. (I don't think we are having a language difficulty about "countable $\ne$ finite", are we?) – fleablood Nov 01 '22 at 06:58
  • In my previous comment, I didn't specify $k$ is an integer. But that's what was intended. – vvg Nov 01 '22 at 07:04
  • I don't understand. Surely ${2,3,5,7,11,........}$ is a countable sequence. (We aren't haven't language translation problems, are we? We both agree that "countable" doesn't mean infinite and that $\mathbb N$ is countable, don't we?)...... Hmmm, I suppose if we had if we had a function $f$ be a real function and we defined $a_x = f(x)$ then we could maybe call ${a_x|x\in \mathbb R}$ as an "uncountable sequence" and if $f$ is not injective than it isn't pair-wise distinct.... but I'd consider that a gross abuse of vocabulary. – fleablood Nov 01 '22 at 07:16
  • Ah, I see. I used uncountable for the sequence of primes when I should have said infinite. – vvg Nov 01 '22 at 07:21