# What is the difference between $\mathbb{N}$ and $\{ \mathbb{N} \}$?

What is the difference between $$\mathbb{N}$$ and $$\{ \mathbb{N} \}$$?

Note that, here $$\mathbb{N}$$ is the set of all natural numbers.

• Please put your question in the question body. Dec 9 '19 at 21:55

$$\mathbb{N}$$ is the set of natural numbers, $$\{\mathbb{N}\}$$ is a set containing one element, namely $$\mathbb{N}$$.

• Thanks. I was confused, because of the uppercase N inside curly brackets. Dec 9 '19 at 2:24

The same way $$\emptyset \neq \{ \emptyset \}$$.

One is a set with no elements, the other a set containing a set of no elements.

So in your example, the left is the set of natural numbers, the right a set containing one element, the set of natural numbers.

As I tell my students, it's the difference between an empty box and a box containing an empty box.

• Thanks 😊. I was confused because of the uppercase letters. Dec 9 '19 at 2:26
• @RajatDash: The letter is just a name. There's no reason why it should matter whether it is an uppercase letter, lowercase letter, Greek letter, Hebrew letter, or Chinese symbol. Dec 9 '19 at 22:00