According to set theory, are $\{\}$ and $\{\{\}\}$ equal?
Asked
Active
Viewed 75 times
0
-
2NO; the first one is empty while the second one has one element: the empty set. – Mauro ALLEGRANZA Jul 07 '17 at 15:11
-
$P(A)="{}\in A"$ is true for one and false for the other. Therefore $\forall x(x\in{}\leftrightarrow x\in{{}})$ cannot be true. – Bettybel Jul 07 '17 at 15:11
-
1We have ${}=\emptyset$, and ${{}}={\emptyset}$. – Dietrich Burde Jul 07 '17 at 15:12
-
3A bag with an empty bag in it is able to be distinguished with an empty bag. – JMoravitz Jul 07 '17 at 15:13
-
@JMoravitz I'll have to use that one! – Theo Bendit Jul 07 '17 at 15:18
2 Answers
5
No. We have that $\{\}$ is the empty set, whereas $\{\{\}\}$ is the set which contains one element: the empty set. So $\{\{\}\}$ is not empty.
Dave
- 13,568
2
No, in fact one definition of the natural numbers has $\{\}$ as 0, $\{\{\}\}=\{0\}$ as 1, $\{\{\},\{\{\}\}\}=\{0,1\}$ as 2, etc.
Akababa
- 3,109