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Just want to confirm so I don't overthink this. When we say "all elements in a list are distinct", that by definition means no 2 elements in the list are equal, right? In other words all elements in a list are distinct iff no 2 elements in the list are equal?

Kindly please let me know.

EDIT: Change "set" to "list". Hopefully that makes my question more clear and not conflict with the specific definition of "sets". I apologize for that mistake

Bob Marley
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  • Yes - otherwise, if there are copies, you'd be talking about a multiset. – Sean Roberson Nov 13 '24 at 21:27
  • Ponder on this: two things are necessarily not equal: if they were equal, there would only comprise one thing, not two. That's how sets, as opposed to multisets work: ${x, x} = {x}$. – Rob Arthan Nov 13 '24 at 21:38
  • @RobArthan Oh right forgot about that. Ok so instead of sets I say "list of elements" would I then be correct? – Bob Marley Nov 13 '24 at 21:41
  • Technically, it is always the case that the elements of a set are distinct (axiom of extensionality), so this wording is a kind of "mathematical colloquialism". What is actually meant is that whatever representation one is using for the elements, different "representational elements" correspond to different elements. For example, the set ${x,y}$ consists of one or two elements according as to whether $x=y$ or $x \neq y.$ But if it is said that "all elements are distinct", then it means that $x \neq y.$ – Dave L. Renfro Nov 13 '24 at 21:43
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    Not really. "all elements in a set are distinct" doesn't make sense and I would advise you to avoid saying it. A sensible statement might be "the elements of a list are pairwise distinct", meaning that if the list is $x_1, x_2, \ldots$, then if $i \neq j$ then $x_i \neq x_j$. – Rob Arthan Nov 13 '24 at 21:44
  • @RobArthan I'm saying "all elements in a list are distinct iff the elements of a list are pairwise distinct" (replacing "set" with "list"). Is that's incorrect? – Bob Marley Nov 13 '24 at 21:47
  • @Rob Arthan: Slightly revised my first sentence in light of your comment. There's still some colloquialism in it, but it's probably good enough for the purposes here. (moments later) I guess my last sentence should be slightly revised also, but too late now! (My $5$ minutes of editing time are used up.) – Dave L. Renfro Nov 13 '24 at 21:47

2 Answers2

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It is about the wording, when you say two elements you are already assuming that they are different. Check the following example. If I write the following set $$ A=\{a\} $$ $a$ is an element of $A$ and so is $a$ (??), but $a=a$. Is there a contradiction here? I have two elements $a$ and $a$ in a set $A$ such that those two elements are equal.

On the other hand, if you mean that two different elements in a set are always different, then it is tautological.

Another example, we have the set $$ A'=\{a,b\} $$ $a$ is an element of $A'$ and so is $b$, can $a=b$? The answer is yes and if it does, then $A'=A$.

Sets do not restrict the multiplicity of elements but do not remember them either way! Last example that you can check for yourself $$ A=\{a\}=\{a,a\}=\{a,a,a\} $$

Francesco Peña-Garcia
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  • Regarding the title of your question, in set theory every element of a set is also a set, so for two elements to be different it means that they as sets are not equal and the equality of sets is defined as the double inclusion $A\subset B\land B\subset A$. – Francesco Peña-Garcia Nov 13 '24 at 22:08
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    Depends who's set theory. – Martin Rattigan Nov 13 '24 at 22:44
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    +1: That's a nice answer, but due to the vagaries of the English language, in modern English, "duplicity" doesn't mean what you think it does $\ddot{\smile}$. "Multiplicity" is the word you are looking for. Apologies for the idiosyncrasies of my mother tongue! – Rob Arthan Nov 14 '24 at 00:01
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To help you be more precise.

A list is a sequence of elements of some Set, indexed by a Indexing Set I.

A sequence of elements of A is a function into A. The domain of the Function is the Indexing Set.

So, if we have A = {a,b} If we have a function f:{0,1} $\rightarrow$ A

s.t. f(0) = a and f(1) = b

<$x_0$,$x_1$> = <a,b> is what you call a "List" ( a special piece of notation, it is in fact just a function)

where $x_n$ = f(n) where n is in the domain of the function.

Remark: Usually, to match our notions of a "list" we want our Indexing Set to be a Well-Ordered Set - This gives us an "ordered List"

To answer your question, you are correct, but how do you express such a property? How would you check it for an infinite list?

If the elements of a list are distinct, we have the following property

$\forall$m,n$\in$I( m $\neq$ n $\rightarrow$ $x_m$ $\neq$ $x_n$)

This forces that every set in the list, is in fact distinct ( different)

So, every set in a list is distinct, if the function which induces the list, is an Injection.

Michael Carey
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