Is the sample space not a set under ZFL?

[scinoob]

I am reading Introduction to Set Theory (Jech & Hrbacek) and in one of the exercises we're asked to prove that the complement of a set is not a set. I get that if it were a set that would imply that "a set of all sets" (the union of the set and its complement, by the axiom of pairing) exists and that leads to paradoxes. However, does that mean that the sample space is not considered a set? I always thought it was a set (and a quick check on Wikipedia confirms that). So, understandably I'm confused.

Any help? Thanks!

 

[micromass]

I am reading Introduction to Set Theory (Jech & Hrbacek) and in one of the exercises we're asked to prove that the complement of a set is not a set. I get that if it were a set that would imply that "a set of all sets" (the union of the set and its complement, by the axiom of pairing) exists and that leads to paradoxes. However, does that mean that the sample space is not considered a set? I always thought it was a set (and a quick check on Wikipedia confirms that). So, understandably I'm confused.

Any help? Thanks!

The issue is that there are two (related) notions of a complement in use here.

When you pick out a sample space, that means that you pick out a set $\Omega$ which is the setting for all set operations. In particular, the complement of $A$ is then defined as $\Omega\setminus A$.

On the other hand, Hrbacek and Jech do not pick out a sample space. Their complement of $A$ is defined as $\{x~\vert~x\notin A\}$ which is not a set.

So you should really be careful which kind of complement you're working with.

 

[scinoob]

Thanks micromass, I wasn't aware there were two different meanings of the term 'complement'. One last thing on this, could you tell me what is meant by 'the state for all set operations'?

 

[micromass]

Thanks micromass, I wasn't aware there were two different meanings of the term 'complement'. One last thing on this, could you tell me what is meant by 'the state for all set operations'?

Can you please give the context?

 

[scinoob]

Oh, it's from your post (second sentence) :)

 

[micromass]

Oh, it's from your post (second sentence) :)

Is it clear now? I used the wrong translation, sorry.

 

[scinoob]

I am not sure. Do you mean that the sample space is the set of all sets for which set operations (like union, subtraction, etc.) can be performed? If so, could you give an example of sets for which these operations are inadmissible?

 

[micromass]

I am not sure. Do you mean that the sample space is the set of all sets for which set operations (like union, subtraction, etc.) can be performed? If so, could you give an example of sets for which these operations are inadmissible?

It's just that when we specify a sample space, then we agree that these will be the only sets we'll work with. Of course, there will be sets outside the sample space, but we find them less important for our purpose.
In particular, the complement is defined as all elements in the sample space not in the set.

 

[WWGD]

Sample space is usually associated to some specific (informal) experiment , and it is a listing of all possible outcomes of the experiment/situation at hand. Maybe the simplest example is that of throwing a coin once and observing what face shows once the coin settles . The sample space is then {H,T}.