Is it possible to prove that -(-A)=A using the concept of set subtraction?

[Kamataat]

I need to prove that -(-A)=A. I guess it's the same as S-(S-A)=A, where S is the space. So is it true, that if x \in S-(S-A) then x \notin S-A?

- Kamataat

 

[Muzza]

I take it -A means the complement of A (with respect to some universe)? The following are equivalent (~ means "not"):

x \in -(-A)
x \notin -A
~(x \in -A)
~(x \notin A)
~(~(x \in A))
x \in A

That establishes the two inclusions -(-A) \subseteq A and A \subseteq -(-A).

 

[Kamataat]

Thanks, Muzza, I get your proof. Still, why are the NOT steps neccessary? Why not this instead:

x \in -(-A)
x \notin -A
x \in A?

If you go (in your post) from step #1 to step #2 directly, then why don't you go from #2 to #6 (e.g. skip #3, #4 and #5)? I mean, if from #1 follows #2, then doesn't #6 follow from #1 and #2 combined (w/o the intermediate steps)?

- Kamataat

 

[Muzza]

*shrug* Do as you please :P

 

[Kamataat]

ok, tnx

- Kamataat