# How does the empty set work? [duplicate]

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I'm having trouble understanding what exactly an empty set is. Does $\varnothing$ mean $\{\}$ ? and what is the difference between $\varnothing$ and $\{\varnothing\}$ ? If someone could shed some light on this, and provide a couple of more examples to help me learn what this weird concept is, I'd greatly appreciate it.

Thanks!

## marked as duplicate by Andrés E. Caicedo, Claude Leibovici, Avitus, Mark Bennet, Ian ColeyApr 8 '14 at 7:23

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• Both questions have been covered extensively on the website before. Did you search on the website, or on Google before posting this question? – Asaf Karagila Apr 8 '14 at 3:15
• Some posts: this and that and also this and of course something like that. There are more. Many more. (Also, I didn't downvote, in case that it might seem like that.) – Asaf Karagila Apr 8 '14 at 3:23
• Of course, if you did read a few of these threads, and you still have questions it would be nice to know that you did in fact search the internet, read a bit, and what is it that you don't fully grasp yet. – Asaf Karagila Apr 8 '14 at 3:24
• The empty set is just what the name implies. It is $\{\}$, the set with no elements. $\{\emptyset \}$ has an element, which is the empty set. – Ross Millikan Apr 8 '14 at 3:39

## 2 Answers

$\emptyset=\{\}$. Now, $\{\emptyset\}=\{\{\}\}$. Thus, $\{\emptyset,\{\emptyset\}\}\ni\emptyset$, as is visible. Now, what is interesting is that we can define a number system: $$0=\emptyset,1=\{\emptyset\},2=\{\emptyset,\{\emptyset\}\},3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},...$$ Thus, we may relate $\emptyset$ to something very well known: the natural numbers $\mathbb{N}$! Hopefully this makes it "concrete", though I admit, it still is very abstract.

It is indeed, as you put it, $\emptyset = \{\}$. Just imagine an empty bag. It has nothing inside. Now, it is not the same as $\{\emptyset\}$, which would be a bag with an empty bag inside. The outer bag is not empty, it contains another bag.

• On a related note, is ∅ an element of {∅, {∅}} ? This stuff is ridiculously abstract. – Vimzy Apr 8 '14 at 3:21
• It is :) $\emptyset$ as an element works just as any other element. If it bothers you too much, just change its name! And just know that it has no items inside. Other than that, it works the same! – SantiagoC Apr 8 '14 at 3:26
• @Vimzy - all of mathematics is abstract, and most of it may look "ridiculously abstract"... but with it the man landed on the moon and your bank every month calculates the "total" of your bank accont. So, can be useful... – Mauro ALLEGRANZA Apr 8 '14 at 8:30