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

Solution 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.

Solution 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\} $.

Solution 3:

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.

Solution 4:

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 $S$.

Solution 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.