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I'm currently reading Book of Proof by Richard Hammack. In the chapter on sets, he gives this as an example:

X = { n2 : n ∈ Z }

If n can be any integer and (-n)2 = (n)2, are the elements of this set distinct? If they are not, is this a set?

To my understanding, sets cannot have repeated elements. However, on the wikipedia page for sets it defines a set as

a collection of distinct objects, considered as an object in its own right

but later says

In an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}.

So can sets have repeated elements?

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Think of it like this: an element $x$ of a set $A$ can be mentioned in the notation of the set more than once.

Here $\{11,6,6\}$ is a notation of the unique set that only has the elements $6$ and $11$ (and is completely determined by that fact) and in this notation element $6$ is mentioned twice. That is allowed.

That's all.

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  • $\begingroup$ So how would I distinguish a set that mentions an element multiple times and a multiset that contains multiple identical objects that are separate elementst? $\endgroup$ – Copeland Corley Dec 4 '18 at 14:36
  • $\begingroup$ By paying attention to the context that you are working in, I would say. Multisets and sets are distinct mathematical objects. Also see here where is mentioned e.g."...to distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset $\{a,a,b\}$ can be denoted as $[a, a, b]$...". I assure you that if you once work with multisets then you will certainly be aware of that too :-). In your question you only mention sets and not multisets. $\endgroup$ – drhab Dec 4 '18 at 14:58
  • $\begingroup$ One final question. Since {11, 6} and {11, 6, 6} are the same set, do they have the same cardinality? $\endgroup$ – Copeland Corley Dec 4 '18 at 15:33
  • $\begingroup$ Yes, they do. The cardinality is $2$ as you will understand. $\endgroup$ – drhab Dec 4 '18 at 16:40
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Yes, it is a set. Note that $\{1,1\}$ and $\{1\}$ are the same set. So$$X=\{1,1,4,4,9,9,16,16,\ldots\}=\{1,4,9,16,\ldots\}$$

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  • $\begingroup$ Okay, so if I have the set {1,1} and the multiset {1,1}, they are not equal, right? $\endgroup$ – Copeland Corley Dec 4 '18 at 14:39
  • $\begingroup$ I have no experience whatsoever with multisets. $\endgroup$ – José Carlos Santos Dec 4 '18 at 14:40
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The notation allows to repeat the elements, but as they are identical, they "count once" only. So, strictly,

$$\{11,6,6\}=\{11,6\}.$$

True replication requires a multiset.

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  • $\begingroup$ How would I know that I am dealing with a set instead of a multiset? Is it supposed to be inferred from the context? $\endgroup$ – Copeland Corley Dec 4 '18 at 14:37
  • $\begingroup$ @CopelandCorley: of course. Multisets aren't so often met. $\endgroup$ – Yves Daoust Dec 4 '18 at 14:55

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