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

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

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  • $\begingroup$ Please put your question in the question body. $\endgroup$
    – celtschk
    Dec 9 '19 at 21:55
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$\mathbb{N}$ is the set of natural numbers, $\{\mathbb{N}\}$ is a set containing one element, namely $\mathbb{N}$.

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  • $\begingroup$ Thanks. I was confused, because of the uppercase N inside curly brackets. $\endgroup$
    – Rajat Dash
    Dec 9 '19 at 2:24
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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.

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  • $\begingroup$ Thanks 😊. I was confused because of the uppercase letters. $\endgroup$
    – Rajat Dash
    Dec 9 '19 at 2:26
  • $\begingroup$ @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. $\endgroup$
    – celtschk
    Dec 9 '19 at 22:00

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