The difference between $\{\mathbb{N}\}$ and $\mathbb{N}$? $\mathbb{N}$ stands for Natural numbers.
What does each mean and how are they different.
Thanks.
 A: Assuming that $N$ here refers to any given set and not specifically the set of Natural Numbers, $\mathbb{N}$, 


*

*$N$ refers to any given set. It is what you define it to be.

*$\{ N \}$ is specifically the set containing $N$, and only, $N$. Because of this characterising property where $\{ N \}$ contains one and only one element, it is known as a singleton. 


The difference becomes especially important when we are discussing the empty set. For example, 


*

*$\emptyset$ is the set containing no elements, i.e. it is empty. 

*$\{ \emptyset \}$ is the set containing the empty set. It is not empty, simply because it contains the empty set $\emptyset$. 

A: $\mathbb{N}$ is a set containing infinitely many elements.  Depending on convention, $0$ may or may not be an element of $\mathbb{N}$.  So either
$$\mathbb{N} = \{ 0, 1, 2, 3, 4, \dots \}$$
or
$$\mathbb{N} = \{ 1, 2, 3, 4, \dots \}$$
But $\{ \mathbb{N} \}$, in contrast, is a set containing a single element, namely $\mathbb{N}$.  So (for example) the number $5$ is an element of $\mathbb{N}$, but it is not an element of $\{ \mathbb{N} \}$.
A: $\Bbb N$ has infinitely many elements. Its elements are $0$, $1$, $2$, etc.
$\{\Bbb N\}$ has only one element. Its element is $\Bbb N$.
