# Is every empty set equal? [duplicate]

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1. In a math textbook I found a statement that every empty set is equal.
2. There are such sets that it is impossible for them not to be empty: set of all natural numbers between 10 and 11.
3. There are such sets that it is possible for them not to be empty: all the people from the Earth, who are on the Mars now. In future it can be possible.
4. Quesstion: can sets mentioned in 2 be equal to sets mentioned in 3. If yes, why not to take in account the difference examples above?

## marked as duplicate by user21820, Asaf Karagila♦Aug 17 '18 at 7:03

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• I think the empty set is one and only one...even if it is subset of every known set in Set Theory – dmtri Aug 17 '18 at 4:54
• Set theory does not by itself deal with sets that can change over time. If you want you can talk about the set of "all the people from the Earth, who are on the Mars on 16 August 2018", which will remain empty in the future. – Rahul Aug 17 '18 at 4:56
• All empty sets are equal because they have the same elements. However, the empty set in (0,2) has a different complement from the empty set in (0,2] – BCLC Aug 17 '18 at 5:47
• Your question is not really about the empty set. You could ask about the set of siblings one might have, which can grow bigger or later shrink smaller. – Asaf Karagila Aug 17 '18 at 7:00
• There's only one empty set. There's only one 1. Do you sometimes ask whether you have "two 1s"? What do you mean by "two" of one thing? Language is being used sloppily/informally. You are interested in something like whether two expressions both evaluate to the empty set. Please quote the textbook. PS Why can't "[expression values that are] sets mentioned in 2 be equal to [expression values that are] sets mentioned in 3"? If it arises, it arises. – philipxy Aug 17 '18 at 7:08

## 2 Answers

Two sets are equal if and only if they contain the same elements. Obviously, there cannot be two different empty sets under this definition. The mistake in your description of three is that you say, that the set of all Earthling who are on Mars now is empty, but in the future there may be Earthlings on Mars. In that case, it will be a different set.

The word "now" is the problem. That isn't sufficiently precise for a mathematical definition. You should make a definition like, "The set of all Earthlings on Mars on August 16, 2018." This set is equal to the set of all integers strictly between $10$ and $11.$ If may happen that the set of all Earthlings on Mars on August 16, 3018 turns out to be nonempty, but that has nothing to do with the case.

• Anno Domini :-) – Arnaud Mortier Aug 17 '18 at 5:20

Every empty set is same in the sense that if you take two empty sets, say $\emptyset_1$ and $\emptyset_2$, then they are contained in one another. You can in fact give a logical argument for this.

If you take any element $x \in \emptyset_1$ (which is none) it is also contained in $\emptyset_2$ and vice - versa. Therefore, $\emptyset_1 = \emptyset_2$. Therefore, we say that all empty sets are equal.

Currently, there are no people on Mars. So, the set mentioned in $3$ is empty. Also, we know that there are no Natural numbers between 10 and 11. So, the set in $2$ is also empty. Now, if you consider the above argument I made, you will understand that these two empty sets are equal even though, the contexts they are made in are different.

• $\emptyset$ $\emptyset$ – Arnaud Mortier Aug 17 '18 at 5:14
• Thank you! I will keep this in mind from next time onwards. Actually, while speaking, I have a habit of speaking "phi" for an empty set. So, the case! – Aniruddha Deshmukh Aug 17 '18 at 5:16
• You're welcome! But I think that "is empty" takes just as much time and is more universally understood than "equals phi" ;) – Arnaud Mortier Aug 17 '18 at 5:18
• $\emptyset_1$ is questionable notation, $\phi_1$ is ok. (In general, $\emptyset$ is of course the right choice.) – Carsten S Aug 17 '18 at 6:59
• Two empty sets are (non-strict) subsets of each other, they are not contained in each other. – immibis Aug 17 '18 at 7:48