A set with members allowed to appear more than once

Question

I'm looking for a definition for a set which its members could be appeared more than once! for example: $$D=\{1,1,2,4,6,6\}$$ Could we call this a set?

Answer 1

This object is variously called a "multiset" or a "bag". It is not, however, a set. Sets are characterized by the $\in$ relation:

$$A=B $$

is defined to mean that $$x\in A\iff x\in B$$

for every $x$. So your object is a set, but considered as a set, it is equal to the set $\{1,2,4,6\}$ and also to the set $\{1,1,2,2,4,4,6,6,6\}$, which I gather is not what you want. If you want your object to be different from $\{1,2,4,6\}$, you have to do something new.

Instead of inventing a new type of object, mathematicians often model it as a function mapping from the domain of elements to the non-negative integers: for each possible element $e$, the function says how many times element $e$ appears in the multiset. Your example $D$ would be modeled by the following function:

$$\begin{align} D(1) & =2\\ D(2) & = 1\\ D(4) & = 1\\ D(6) & = 2 \\ D(n) & = 0\qquad\text{for any other $n$} \end{align}$$

This paper, "Multiset theory", by Wayne D. Blizard, has an axiomatic treatment of multisets and a fairly detailed description of the considerations involved in dealing with multisets.

Answer 2

It looks like the "word" you're looking for is "Multiset":

From Wikipedia (entry linked):

In mathematics, the notion of multiset (or "bag") is a generalization of the notion of set in which members are allowed to appear more than once. For example, there is a unique set that contains the elements a and b and no others, but there are many multisets with this property, such as the multiset that contains two copies of a and one of b or the multiset that contains three copies of both a and b. The term "multiset" was coined by Nicolaas Govert de Bruijn in the 1970s. [Boldface mine.]

Historically:

The use of multisets in mathematics and beyond predates the name "multiset" by many centuries: Knuth (1998) attributes the first study of multisets to the Indian mathematician Bhascara Acharya (circa 1150), who described permutations of multisets.

• Knuth, Donald E. (1998). The Art of Computer Programming – Vol. 2: Seminumerical Algorithms (3rd edition ed.). Addison Wesley. p. 694. Knuth also lists other names that were proposed for multisets, such as list, bunch, bag, heap, sample, weighted set, collection, and suite.

Answer 3

A multiset which can, funnily enough, be written as a set. Or rather, it can be written as a pair $M=(A,f)$ where $A$ is the underlying set of elements that appear in $M$ and $f\colon A\rightarrow\mathbb{N}_{\geq 1}$ sends an element in $a\in A$ to the number $n\in\mathbb{N}_{\geq 1}$ of how many times $a$ appears in $M$.

The various ways of writing functions and pairs as sets then turns a multiset in to a set.