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In Linear Algebra Done Right, Axler says that order and repetiton don't matter in a set. For example, the set {4} = {4,4} = {4,4,4} or {1,3} = {3,1}. My question is then what do these sets mean. What could {4,4} possibly represent in real life and why would that be equal to what {4,4,4} represents? And the same question for what {3,1} could possibly represent and why that would be equal to what {1,3} represents.

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    $\begingroup$ If I have a bag containing an apple and an orange, that could be represented by $\{\text{apple}, \text{orange}\}$. That bag also contains an orange and an apple, so it could be $\{\text{orange}, \text{apple}\}$. Of course, it's the same bag and the same contents.... $\endgroup$ – Randall May 22 at 1:35
  • $\begingroup$ For the $\{4,4,4,\dots\}$, consider the set as a bag that always contains $4$. Every time you take an item out of the bag, you get $4$. $\endgroup$ – abiessu May 22 at 1:38
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    $\begingroup$ It is harder to give a "real-life" reason for $\{4\} = \{4,4\}$. It's just silly (gives no extra information) to list the same thing twice... $\endgroup$ – Randall May 22 at 1:38
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    $\begingroup$ If I list the set of all people who have ever won a major golf championship, Tiger Woods only belongs on the list once, even though if I enumerated the various wins he'd end up in $14$ different positions on the list. $\endgroup$ – Robert Shore May 22 at 1:50
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If you want a "real life" analogy, consider a shopping list. The order of the items doesn't really matter, and you might have written some of them down several times. So

apples
oranges
milk

is the same as

milk
oranges
apples
milk

It's convenient to have a data structure like this in real life. If I want to make sure to get milk, then I just write it at the bottom of the list; the list doesn't "break" if milk is already there, and I don't care that it's not in alphabetical order.

Sets in mathematics are convenient for exactly the same reason.

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You would never write the set $\{2,3,3,4,4,4,5,5,6\}$ in real (mathematical) life but you might construct it inadvertently. For example, $$ \{ n \quad | \quad n = x+y \text{ where } 1 \le x,y \le 3 \}. $$ That set has just five elements.

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Sets are governed by axioms and the most basic one is the extensionnality axiom according to which :

set A and set B are equal (identical) iff x belongs to A iff x belongs to B

or, if you prefer,

set A and set B are equal iff : there is not object x such that x belongs to A and does not belong to B , and reciprocally.

Now, can you see an object that belongs to {4,4} and that does not belong to {4}? and can you see an object that belongs to {4} and that does not belong to {4,4}?

If you answered "no" to both questions, you know that the set {4,4} and the set {4} are exactly the same set, they are equal.

You may ask yourself the same two questions regarding the sets {3,1} and the set {1,3}.

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Set theory at first, may not look like it is very useful. Strangely enough it has diverse applications, even in Relational Database Systems. In fact the entire Theory of RDBMS is based on the set theory which I found strange when I studied it.

I used to think that set theory is another abstract topic in mathematics but I was surprised to find its role in databases.

Some real life examples to answer your questions.

Order does not matter in a set. This is a "definition". If you'd care about an order, another mathematical object like a "List" may work for you. As an example, say you have 2 girl friends Magy and Megan. we could say that your set of girl friends is {Magy, Megan} or we could say {Megan, Magy}. The order does not matter. This is also true if you draw a Venn diagram. You can lay the elements in any way you like for one set.

Repetition does not matter in Multisets.

What do sets like {4,4,4} and {4,4} mean? We can't guess the meaning from just the set in this way. It would make since though to say that the set of all solutions to the equation $(x-1)(x-2)(x-3)=0$ is the set of values S={x=1, x=2, x=3}. This is meaningful. Here also, order does not matter.

As for your question, "for what {3,1} could possibly represent and why that would be equal to what {1,3} represents." these $2$ sets may be the solution of some equation such as $(x-1)(x-3)=0$. They are equal, because order does not matter and because they are the same values that satisfy the condition.

It is important too keep in mind that a lot of things in this world is based on the definitions we give to things. Understanding the definition will make understanding the following concepts easier.

Hope this helps a bit.

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Think of a lottery.

You can mark 4 as often as you want, but when the cross gets bolder this does not mean that 4 counts multiple times. In addition, the order of the crossed numbers does not matter.

The nice thing about the analogy is, that variants of the lottery can be compared to variants of sets, e.g., as multisets or ordered sets, for gamblings in which you actually may have more than one 4 or where the order of the drawn numbers is important.

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