It makes intuitive sense to me, but I am not sure how to prove it.
The exact wording of the exercise is:
Let X be a set, and let P(X) denote its power set. Show:
- (P(X), ⊂) is an ordered set.
- (P(X), ⊂) is a totally ordered set if and only if X = ∅ or X = {a}.
If I know how to do 1., I think I would be able to do 2. alone without help.
I would know what to do if it were (P(X), ⊆) but I am confused, because:
It is not reflexive since A ⊂ A for all A ⊂ P(X) is not true because of A = A.
I am not sure if it is antisymmetric since the definitions I found would not work because of similar reasons as above.
I do know it is transitive, however.
So what type of ordered set would this be, and how would I prove it? I would appreciate the help.