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Consider the following statement: ∅∈{0}. Is this statement is true or false? I can't determine the answer to this question as I am confused. The "∈" states that certain element belongs to a certain set ( for example a ∈ A) . But in the question, we are saying that an empty set ( not an element) belongs to (∈) another set ({0}) . So I am confused here, is this statement true or false?

Source : Discrete mathematics and its Applications 8th edition (Kenneth Rosen) Excercise 2.1 Sets -- Question 11 part(b)

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    What is "0"? In set theory, it is often defined to be the empty set, in which case the statement would be true. – Arturo Magidin Oct 18 '21 at 03:28
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    @ArturoMagidin Correct me if I am wrong, but while 0 is typically defined to be the empty set, is it not still distinct from ${0}$? – JJ Hoo Oct 18 '21 at 03:29
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    @ArturoMagidin The empty set is not the same as the set containing the empty set – JJ Hoo Oct 18 '21 at 03:31
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    @JJHoo So? If zero is the empty set, then the statement reads $\varnothing\in{\varnothing}$, which is true. – Arturo Magidin Oct 18 '21 at 03:32
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    One point of confusion that you seem to have is that you seem to think there are two different types of objects: sets and elements. But sets can be elements of other sets. For one example, look up the definition of the "power set" of a set. – Joe Oct 18 '21 at 03:33
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    Is your question supposed to have two different symbols, $\emptyset \in {0}$, or the same symbol twice: $\emptyset \in { \emptyset }$? If two, can you tell us what each symbol is? – Joe Oct 18 '21 at 03:37

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It seems unlikely to me that $\emptyset$ and $0$ should represent the same thing in context. Assuming that they are different, the statement is incorrect. While it is true that the empty set is ALWAYS a subset of any other set, it is not necessary that it is an element. The empty set only belongs to a set if it is actually written as an element when defining the set. Some more details can be found in this question.

I encourage you to also think about the following question:

Is the empty set always an element of the power set of any set?

Now, as noted in the comments, it is sometimes the case that the symbol 0 is used to denote $\emptyset$. If this has been explicitly defined in your references, then you indeed have that this statement is true, by definition.

JJ Hoo
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    The natural number/ordinal 0 is often defined as the empty set. In which case the statement woulf not be incorrect. Its correctness depends on the precise set theoretic meaning of "0". – Arturo Magidin Oct 18 '21 at 03:29
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    I suspect that the two symbols were intended to be the same, but didn't render that way. – Joe Oct 18 '21 at 03:30
  • @ArturoMagidin I agree that 0 is often defined as the empty set, but as I mentioned in another comment, the empty set is not the same as the set containing the empty set. In fact, some construct the natural numbers using sets containing empty sets inductively. – JJ Hoo Oct 18 '21 at 03:32
  • Why are you talking about equality? The statement is about membership, not equality. – Arturo Magidin Oct 18 '21 at 03:33
  • I see your point now, thank you for the clarification on that. However, without further clarity, I believe it disingenuous to assume that if $0$ were indeed used to define the empty set, the symbol $\emptyset$ would still be used. – JJ Hoo Oct 18 '21 at 03:35
  • My ultimate point is that there is not enough clarity in the statement, so answering with a categorical "This is incorrect", which seems to unwarranted and jumping the gun. Right now, without clarification, it seems to me the question is not well-formed, but your are asserting it is definitely, without any question, "incorrect." – Arturo Magidin Oct 18 '21 at 03:44
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    I see what you are saying, thank you for the clarification, I shall edit my post to reflect this! @ArturoMagidin . Apologies for my ignorance. – JJ Hoo Oct 18 '21 at 03:46
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$\emptyset \in \{0\}$ if and only if $\emptyset = 0$.

Whether this is true depends on which $0$ you're talking about.

Assuming you're referring to the normal definitions of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, etc.,

  • $0 \in \mathbb{N}$: $0$ is the empty set
  • $0 \in \mathbb{Z}, \mathbb{Q}, \mathbb{R}$, or $\mathbb{C}$: $0$ is not the empty set
Mark Saving
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    I think your first bulleted point only adds confusion for the OP. $\mathbb N$ is not universally taught to exclude the number $0$. – amWhy Oct 18 '21 at 20:55
  • @amWhy I'm not sure what you mean. I simply stated that if you're referring to the natural number $0 \in \mathbb{N}$ under the typical set-theoretic definition of $\mathbb{N}$ (as the set of all finite orderinals), then $0$ is the empty set. Obviously $0$ is an element of $\mathbb{N}$. – Mark Saving Oct 18 '21 at 20:58