What is the difference between $x$ and $\{x\}$ when $x$ itself is a set?

Question

I've already asked this a part of another question, but thought it'd be easier to elaborate a bit more on my concern.

Let $x$ be a set. What is the difference between $x$ and $\{x\}$? I get that the latter is a set consisting of a single element - namely $x$, but what is the difference?

For example, we can have $x$ to be the set $\{1\}$, then $\{x\}=\{\{1\}\}$. Aren't those $2$ expressions the same?

Another problem are the brackets - when we have a set, do we always have to surround him with brackets, for instance, can we have $x$ to be the set $2$?

Thanks a lot

Do you understand the distinction between the emptyset and the set containing the emptyset (${}$ versus ${{}}$)? — Noah Schweber, Jan 25 '18 at 13:36
Yes, the former contains nothing, while the latter contains the empty set — asdf, Jan 25 '18 at 13:36
Les trivial example: $\mathbb N = { 0,1,\ldots }$ has infinite many elements: all the natural numbers, while ${ \mathbb N }$ has only one element: the set of all natural numbers. — Mauro ALLEGRANZA, Jan 25 '18 at 13:38
So then why do you think that ${1}={{1}}$? That would only be true if $1={1}$, and what's your reason for believing that? — Noah Schweber, Jan 25 '18 at 13:38
You may find John von Neumann's construction of the ordinals interesting. We start by defining $0={}$, then $1={0} = {{}}$, $2 = {0, 1}$, $3 = {0, 1, 2}$, etc. — PM 2Ring, Jan 25 '18 at 13:42
I see, I didn't see the difference between sets and elements grouped together — asdf, Jan 25 '18 at 13:43
"I see, I didn't see the difference between sets and elements grouped together" -- "elements grouped together" are not a mathematical object. A set is. You can't say that "$a,b,c$" is anything mathematically, because it's not a single object. But ${a,b,c}$ is an object, and you can talk about its properties. — Deusovi, Jan 25 '18 at 21:58
One of the most popular mathematical "trivia", even among people not having much common with math, an analogy to something obvious or trivial, is saying that $1+1=2$. Maybe we could popularize that $1+1={1}$, which looks even better for me :-) — Kusavil, Jan 26 '18 at 03:38

score 33 · Accepted Answer · answered Jan 25 '18 at 13:38

33

Think of the brackets as a bag you put things in. Then $\{1\}$ is a bag containing the number $1$. But $\{\{1\}\}$ is a bag containing a bag containing the number $1$. So two bags, one inside the other. These are different. Physically different if you think real paper bags.

answered Jan 25 '18 at 13:38

Ethan Bolker

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To the OP: note that this is a limited intuition. In particular, it's really more about multisets than sets, since a bag containing two empty bags is different from a bag containing one empty bag. Still, it's quite useful for thinking about this sort of issue at the beginning. – Noah Schweber Jan 25 '18 at 13:39
Thanks a lot! Really appreciate this intuitive definition – asdf Jan 25 '18 at 13:43
@NoahSchweber Thanks for the clarification. – Ethan Bolker Jan 25 '18 at 14:57
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${}$ and ${{}}$ is more fun; one has cardinality 0, the other 1. – Yakk Jan 25 '18 at 16:37

score 4 · Answer 2 · answered Jan 25 '18 at 13:35

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$$\{1\} $$ is a set whose the unique element is the integer $1$

$$\{\{1\}\} $$ is a set whose the unique element is the set $\{1\} $.

answered Jan 25 '18 at 13:35

hamam_Abdallah

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I see, thanks! But what about the brackets bit - do we have to always put them around a set to indicate that it's actually a set and not, say, some integers? – asdf Jan 25 '18 at 13:38
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@asdf In response I would ask you, what do you consider to be the difference between "a set" and "some integers"? – David Z Jan 25 '18 at 21:13
@DavidZ it is clear there is a difference between an element and a set. – hamam_Abdallah Jan 25 '18 at 21:45
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It is clear to most of us here, yes, but I'm specifically asking @asdf. – David Z Jan 25 '18 at 21:47
Yes, that was my problem - I basically made no difference, thank you all – asdf Jan 25 '18 at 21:51
@Salahamam_Fatima Now I wonder, what the difference really is? I mean, every element possible is a set too, right? (we start from defining integers like John von Neumann in his construction, then we build "the rest" of sets and mathematics). And every set $X$ is always element of some set, for example of ${X}$ or its own power set. So, aside of differentiating a set from his own element in these special cases, can we say that, in general, there is any difference between sets and elements? – Kusavil Jan 26 '18 at 03:49
In modern mathematics, everything is a set! So yes, 2 is a set and it is { {}, {{}}}. For natural numbers, the set is defined by 0={} (emptyset), then n+1=n cup {n}. Computing a few terms can clarify the situation. – Andrea Marino Jan 26 '18 at 11:53

asdq · Answer 3 · 2018-01-27T08:24:16.470

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Well if you have $x=\varnothing$, then $0=\#x\neq \#\{x\}=1$. So clearly both sets are not the same.

Edit: With $\#S$ I refer to the cardinality of a set $S$, i.e. in the finite case the number of elements in $S$.

edited Jan 27 '18 at 08:24

answered Jan 25 '18 at 13:37

asdq

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Here $#$ is being used to denote "the cardinality of" a set, i.e. the size of the set in terms of how many elements it contains. – hardmath Jan 25 '18 at 13:41
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@hardmath Thank you, I should have explained the notation. – asdq Jan 25 '18 at 13:42
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@asdq You could still edit your answer to explain the notation. – David Z Jan 25 '18 at 18:44
@DavidZ Since it is explained right below, I don't see the benefit. – asdq Jan 25 '18 at 22:29
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@asdq On StackExchange comments are temporary. You shouldn't rely on comments sticking around. It's explicitly part of the StackExchange usage guidelines that improvements suggested in comments should be incorporated into the question/answer itself. – R.M. Jan 26 '18 at 00:09
@R.M. I see, I was not aware of that. – asdq Jan 27 '18 at 08:24

spkakkar · Answer 4 · 2018-01-25T19:39:28.427

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You are probably getting confused between the name of a set and its description.
When we write $A=\{x\}$, we mean $A$ is a set and inside set $A$, we have an element $x$.

Now if I define another set $B=\{A\}$, then $B$ is a set and inside set $B$, we have an element $A$, which is also a set. In this case, $B$ is a set of sets.

If you want to refer to the latter set, write
its name $B$, or
its description $\{A\}$.

For your last question, YES, we surround the elements of the set by curly braces {}, which also ensures unorderdness and non-repeatability.

edited Jan 25 '18 at 19:39

answered Jan 25 '18 at 13:45

spkakkar

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I'd suggest rephrasing to make it clear that ${B}$ is a perfectly fine thing to write, it's just an entirely different mathematical object from $B$. – David Z Jan 25 '18 at 18:45
Thanks for suggestion. I have removed dos and donts. – spkakkar Jan 25 '18 at 19:41
After the edit, I think this is a really nice way of describing it. – David Z Jan 25 '18 at 21:11

score 1 · Answer 5 · answered Jan 25 '18 at 14:42

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There is a practical difference when you think about how you might use these sets - namely as a domain of functions. A function that takes a number is not the same as a function that takes a set.

answered Jan 25 '18 at 14:42

Kevin Olree

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score 1 · Answer 6 · answered Jan 25 '18 at 15:17

Perhaps it would be helpful to imagine the difference in concrete terms - say in terms of a computer data structure. Suppose we represent sets using linked lists [disregard for the moment that we cannot represent infinite sets this way]. Then $x$ is (the head pointer of) a sequence of nodes, where each node has a pointer to an element of $x$ and a pointer to the next node in the sequence. Then {$x$} is (the head pointer of) a single node, whose element pointer points to (the head node of) the sequence of nodes representing $x$. Clearer?

What is the difference between $x$ and $\{x\}$ when $x$ itself is a set?

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