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I know that (I might be wrong):

  • Symbol for empty or null set : {Ø} or {}
  • Null or empty set is 'subset of all sets' as well as 'empty or null set' set
  • So, { {} } is same as { Ø }

I just want to know { {} } or { Ø } is an empty set or not ? And if yes then we can conclude that if a set contains a only null set which is by definition always true then it must be a null or empty set.

(Here I am assuming empty and null are same, because I've read that they sometimes taken as different.)

Andrés E. Caicedo
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Sumeet
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  • Does this answer your question? Empty set does not belong to empty set – Alex Dec 17 '19 at 17:30

  • Symbol for empty or null set : ${Ø}$or ${}$

    this is wrong. It should be "symbol for the empty set is $Ø$ or ${}$

    Empty set is a subset of all set by definition of a subset.

    we say $A$ is a subset of $B$ if some element $a$ belongs to $ A$ then $a$ belongs to $B$.

    Now since empty set contains no elements the above statement is vacuously true. But depending on that you cannot write empty set as ${Ø}$ or ${{}}$

     – Prabath Hewasundara Dec 17 '19 at 17:52
  • ${Ø}$ and ${{}}$ is equal to the set which contains empty set. In modern-day we do not use the word "set" for these kind of sets. sets which contains sets are called collection of sets , family of sets. – Prabath Hewasundara Dec 17 '19 at 17:52

4 Answers4

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A set whose only element is the empty set is not empty (an empty set contains no element).

Think of sets a boxes. If you put a small empty box into a big box, the big box isn't empty anymore. It doesn't matter if the small box is empty or not. That's the beauty of the $\{\;\}$ notation -- it "looks" like a box.

If you remember that $\varnothing$ is just another name for $\{\;\}$, then you immediately know that $\varnothing$ and $\{\varnothing\}$ are not the same thing, but $\{\varnothing\}$ and $\{\{\;\}\}$ are the same thing.

MPW
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I know that (I might be wrong):

  • Symbol for empty or null set : {Ø} or {}

You write the empty set as "Ø" or "{}" so your first notation, "{Ø}" is already a set containing the empty set!

So don't mix:

  • the empty set: "Ø" or "{}"
  • a set containing the empty set: "{Ø}" or "{ {} }"

So, { {} } is same as { Ø }

Correct!

Community
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StackTD
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Yes they are the same and it is a set witch contains an empty set as an element.

But symbols {Ø} and {} are not the same. Second one is an empty set or $\emptyset$

nonuser
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The sets you have written down are the same. Neither is empty since each has just one element: the empty set.

Related: What is the difference between $x$ and $\{x\}$ when $x$ itself is a set?

Ethan Bolker
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  • "The sets you have written down are the same" — there is quite a number of sets written in the question, not all being the same. – lisyarus Dec 17 '19 at 17:33
  • @lisyarus Before the edit there were just two sets listed - the two in the paragraph beginning "I just ..." – Ethan Bolker Dec 17 '19 at 17:44