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Is it possible that there exist a set $U$ that has the property $U=U^c$?

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    $\begingroup$ I guess if the universal set is the empty set. $\endgroup$
    – IAmNoOne
    Commented Nov 16, 2015 at 1:46
  • $\begingroup$ By the principle of extensionality, two sets are equal when they contain the same elements. Can the complement of a set $U$ contain the same elements as $U$ does? $\endgroup$
    – hardmath
    Commented Nov 16, 2015 at 1:46
  • $\begingroup$ @Nameless: Wouldn't the universal set have to contain the empty set? So in most understandings the universal set is not empty. $\endgroup$
    – hardmath
    Commented Nov 16, 2015 at 1:48
  • $\begingroup$ @hardmath, I guess that's true, but every set contains itself anyways. $\endgroup$
    – IAmNoOne
    Commented Nov 16, 2015 at 1:50
  • $\begingroup$ @Nameless: I'm distinguishing the relationship of a set containing its elements from that of a set containing subsets. $\endgroup$
    – hardmath
    Commented Nov 16, 2015 at 1:51

1 Answer 1

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This depends on the universal set in question.

For a particular universe, $\Omega$, we define the complement of a subset $A$ of $\Omega$ as:

$A^c = \Omega\setminus A=\{x~:~x\in \Omega~\text{and}~x\notin A\}$

In the case that $\Omega$ is nonempty, then there is some $x\in \Omega$. This implies that exactly one of the following is true: $x\in A$ or $x\in A^c$. Since it cannot be that $x$ is in both, we see that $A\neq A^c$.

If you allow for your universal set to be empty, I.e. in the case that $\Omega=\emptyset$, then the only subset of $\Omega$ is the empty set.

You have in that case $\emptyset^c = \emptyset\setminus \emptyset = \emptyset$

Because this is a particularly uninteresting scenario and provides difficulties for definitions in measure theory and probability, we usually do not allow our universal sets to be empty.

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  • $\begingroup$ Yeah, but it may be the interesting scenario $\endgroup$
    – user
    Commented Nov 16, 2015 at 1:57
  • $\begingroup$ It depends I guess how you define empty $\endgroup$
    – user
    Commented Nov 16, 2015 at 1:59
  • $\begingroup$ There is only one definition of an empty set, and that is the set with no elements. I.e. $\emptyset = \{~\}$ and satisfies the property $\forall x\in \Omega, x\notin \emptyset$. $\endgroup$
    – JMoravitz
    Commented Nov 16, 2015 at 2:01
  • $\begingroup$ Every element in the universe is in the universe. However, there can be elements not in the universe. You cannot have a set of all sets. That leads to Russel's Paradox. $\endgroup$
    – JMoravitz
    Commented Nov 16, 2015 at 2:06
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    $\begingroup$ Read en.wikipedia.org/wiki/Empty_set or go to philosophy.stackexchange.com if you still have concerns. $\endgroup$
    – JMoravitz
    Commented Nov 16, 2015 at 2:13

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