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From the Wikipedia, the set is defined as:

a collection of distinct objects

What is the single word for a collection opposite to this one? The collection of not distinct objects - it means at least 2 should appear. A collection without orphans?

The following ones are qualified because all the items occur more than 2 times:

cat, dog, cat, bird, dog, cat, bird, dog, cat

1, 1, 2, 2, 3, 3

The following ones are not (a dog and 3 occur only once):

cat, dog, cat, bird, cat, bird, cat

1, 1, 2, 2, 3

  • I have no idea what you mean. Can you try explaining a bit more? – FumbleFingers Reinstate Monica Jul 14 '18 at 16:32
  • It looks like what you want is a multiset that's defined in the described manner. I don't think there's a separate word for what you're looking for. – userr2684291 Jul 14 '18 at 16:37
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    I'm voting to close this question as off-topic because it's about an obscure domain-specific usage (which might be better addressed on math.SE). – FumbleFingers Reinstate Monica Jul 14 '18 at 16:45
  • That largely depends on the area where you're going to use the word. For example, in mathematics a sequence is defined as an ordered list of real numbers called terms that never stops. For example, the sequence bₙ = (-1)ⁿ looks like this: -1, 1, -1, 1, -1, 1, -1, ... As you can see, it's a sequence whose terms are either 1 or -1 (it alternates between two numbers). And we call that a list. In this case, it's a list that contains infinitely many elements. But, there is really nothing preventing us from defining lists that are finite. – Michael Rybkin Jul 14 '18 at 16:54
  • As alluded to in one answer, I don't believe the word distinct (in the Wikipedia definition of set) means unique; I suspect it just means distinguishable. So, there is really no "opposite" involved here at all. (Confusingly, the definition of multiset says it, "unlike a set, allows for multiple instances for each of its elements"—but the definition of set also says "a set member can be listed two or more times, for example, {11, 6, 6}.") – Jason Bassford Supports Monica Jul 15 '18 at 3:42
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There is no single word for this.

The definition you give is based off the mathematical definition, in which a set is intended to be the simplest kind of mathematical structure: an object is either in the set, or not. There is no added complexity of "how many times an object is in the set" or any kind of relationship between objects in the set. The English word "set" is here used as a convenient translation of the original German word Menge.

(In formal mathematics the situation is different, "set" is an undefined term and it, along with the undefined expression "is a member of" are used in some axioms such as "the emptyset is a set" and "the union of two sets is a set". Anything theory that obeys these axioms is a set theory.)

Mathematically it would be possible to define a structure like you describe. As far as I know nobody has found a need to do so, and so no such structure has been named.

(The mathematical structure would be a mapping f from some underlying set A to the set {n \in Z | n>1} where f(x) represents the number of times each item occurs. This is a trivial modification of the definition of a "multiset")

In common use "set" means something different: It is a collection of objects that are to be used together in some way: We talk about "A set of golf clubs" or "A set of fine china teacups" There is no particular requirement that the objects be different from each other, indeed you would expect all the teacups to look the same.

We haven't needed a word with the exact meaning you specify, and I believe none exists in English. However if you are looking for a class name in a programming language multiset would do fine.

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A collection of homogeneous objects could be the members of a set (cats).

A collection of heterogeneous objects could also be members of a set (cats, dogs and rabbits).

Homogeneous= all the same Heterogeneous-all different

I have an issue with distinct, which I believe in the definition means individual.

  • To the extent that the word is being used in some mathematical contexts, "distinct" means "distinguishable." For example: "The number of distinct elements in the set {6, 11, 17, 11} is three." – Jeff Morrow Jul 17 '18 at 14:19
  • Well, I distinguish two occurrences of the number 11, separate or distinct from each other.... – Lambie Jul 17 '18 at 14:34
  • To a mathematician, 11 is simply one concept. Repeating its name does not change its singularity. Therefore you cannot distinguish between one eleven and another because there is only one concept. The language of mathematics is highly formalized and artificial. – Jeff Morrow Jul 17 '18 at 16:27
  • @JeffMorrow So, if a set has 100 elements and 50 are the number 1 and 50 are the number two, there are only 2 distinguishable objects? Somehow that does not seem right. – Lambie Jul 17 '18 at 17:14
  • Lambie, all I am saying is that mathematical set theory has its own language. You would not say, under that theory's definitions, that your example set has 100 members; you would say that the set has 2 members. There is even a set that has no members, the null set. – Jeff Morrow Jul 17 '18 at 22:03

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