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

Answer 1

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.

Answer 2

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

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

Answer 3

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 $S$.

Answer 4

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.

Answer 5

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.

Answer 6

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?