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- In a math textbook I found a statement that every empty set is equal.
- There are such sets that it is impossible for them not to be empty: set of all natural numbers between 10 and 11.
- 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.
- 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?
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.
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.
