Set theory: how is $\left\lbrace 1, 1 \right\rbrace$ different from $\left\lbrace 1 \right\rbrace$?

Question

What set theory differentiate sets: $\left\lbrace 1, 1 \right\rbrace$ from $\left\lbrace 1 \right\rbrace$; order does not matter?

Thank you in advance.

Addition/clarification: Thank you very much, everybody, for the providing idea of multi sets - I have found few papers and I hope it will be useful.

I am from microscopy/Imaging field without formal mathematical education, and I try to find the most basic mathematical object which can represent image. I was thinking that set would be the most basic/universal. It works well for just one measurement (or pixel), e.g. {1}; if we have two measurements (pixels), we can have two identical values {1, 1} from set of rational numbers. In classical set theory, it immediately collapses to {1}, so I lose my information. Practically it can happen if I measure number of photons from fixed point inside my object (cell) and I do not care about order of the measurements.

The idea of set is quite attractive as I can embed it in other sets, form universes, etc. On other side, I feel that image always requires some sort of order - except from the very basic measurements I have mentioned before. It can be ordered pair of set of rational numbers and structures, e.g. Cumulative hierarchy which represent pixels (or measurements) and order of the image. I am not sure how easy I can work with them.

Meanwhile I appreciate all your answers, comments and suggestions.

Answer 1

People often refer to these sorts of objects (containers where the order of the elements within them doesn't matter, just how many times each appears) as "multisets".

From any theory of ordinary sets you get a corresponding theory of multisets, and vice versa; an ordinary set can be thought of as just a multiset where nothing appears more than once, and conversely, a multiset can be thought of as just an ordinary set along with a function from that ordinary set to positive cardinalities (representing the number of repetitions of each element).

Answer 2

In elementary set theory, we do not distinguish $\{1,1\}$ from $\{1\}$, we simply characterize a set by the elements it contains. In a similar vein, we say that $\{1,2\} = \{2,1\}$. In general, two sets are equal (by definition) if and only if they contain the same elements, i.e., $A=B \iff \left[x \in A \iff x\in B\right]$. I'll leave it to you to verify that this condition is true for the above two examples.

Answer 3

These objects are often called "bags" or "multi-sets".

The typical method for dealing with bags mathematically is to model them as positive integer-valued functions: the domain of the function is the (ordinary) set of elements actually present in the bag, and the values of the function tell you how many copies of that element (sometimes called its multiplicity) are in the bag.

For example, the two bags you presumably mean to specify in your opening post are:

• The function with domain $\{ 1 \}$ that assigns the value $1 \mapsto 2$
• The function with domain $\{ 1 \}$ that assigns the value $1 \mapsto 1$

Conventionally, we would also allow evaluating such functions outside of its domain, and the value would be zero.

Alternatively, if you have a fixed domain, you could use bags that are nonnegative-valued functions on that domain.

An obvious generalization is to allow more general "number" systems than the positive integers. For example, you could have integer-valued bags or cardinal number-valued bags. Some applications might even want complex number-valued bags!

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It is common to manipulate bags in terms of integer operations, since such formulas are clear and precise. For example, if $f$ and $g$ are functions expressing two bags, then it's obvious what $f+g$ and $\max(f, g)$ would mean: simply apply the specified operation pointwise.

($x \vee y$ is a common notation for $\max(x,y)$ when an infix operator is required, although it should be explained the first time it is used in any source)

However the notion of "union" is unclear — some people have in mind the operation $f+g$ and other people have in mind the operation $\max(f,g)$. Similarly for other operations. The potential advantages of using set-like terminology are generally not worth this risk of confusion when there are perfectly good arithmetic ways to express the same ideas.

Because integer-valued functions are well-studied, it is rather rare to develop a theory of bags independently.