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This is a follow-up to my earlier post regarding the difference between an empty set and a set containing only the empty set. I have done some web research and one analogy that was drawn was a box containing an empty bag. Obviously the box is not empty, because it contains an empty bag. Supposedly the box is like {∅} and the empty bag is like ∅.

Now suppose the box contains an empty coin bag, an empty cookie bag and an empty pencil bag. Can we say that the set contains three null sets?

Mauro ALLEGRANZA
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Sandeep
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    No. An empty coin bag, an empty cookie bag and an empty pencil bag are three different objects. On the other hand, there are no two different empty sets. The empty set is unique, one and only! Thus, the analogy breaks here. –  Sep 22 '21 at 10:20
  • An empty bag is an empty bag. It becomes a cookie bag only when you put some (one or more) cookies in it. – Sayan Dutta Sep 22 '21 at 10:20
  • And ... more generally ... a set cannot contain three "same" objects. Whatever object you can think of (call it $a$), the set with two $a$'s does not exist. ${a, a}$ is just a different way of writing ${a}$. –  Sep 22 '21 at 10:26
  • The issue is: have you understood why the empty set is not the same as the set containing the empty set? Is it clear to you the basic "intuition" about sets, that they are "made of" their elements (and nothing more)? If so, can you compare a set with no element with a set with exactly one element? – Mauro ALLEGRANZA Sep 22 '21 at 10:28
  • In mathematics, sets are "abstract" objects and not physical ones. Thus, using the basic "intuition" above: a set is made of its elements, we have that all the empty collections are the same, because they have no elements, and thus we cannot find a way to discriminate between them. Thus, all "empty sets" are the same empty set, and the first basic result about empty set (following its definition: a set with no elements) is that there is only one empty set and thus we can call it the empty set. – Mauro ALLEGRANZA Sep 22 '21 at 10:31
  • As a tangential remark, a "null set" also has a specific meaning in measure theory (and through that in set theory), which is not necessarily an empty set. So, you may want to avoid the term "null set" for $\varnothing$ to prevent confusion. – Vsotvep Sep 22 '21 at 11:20
  • According to the axiom of extensionality, two sets are equal iff they have the same elements, which holds vacuously for any two empty sets. Therefore, if you posit two or more sets which all have no elements they must all be equal. – Robearz Sep 22 '21 at 16:17

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No, you cannot say that a set has three null sets. It's like saying if a set has elements {$3,3,3,3,3,3$}, the set has $6$ elements, when in reality it only has $1$ element.

MathGeek
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