Are there any sets A for which is totally ordered? Prove your answer.
To be courteous, I will include the definitions for partial ordering and total ordering.
A relation is a partial order if the relation is reflexive, antisymmetric, and transitive. (in this case, the notation just denotes that under the relation is a partially ordered set.
A partially ordered set A with partial order is said to be totally ordered if given any two elements a and b in A, either or .
So, to attempt this problem. I tried making up examples first. The only set A that I could come up with for which is totally ordered is a set with one element. Just to see this, let A = {1}. In this case, { , {1}}. So, if I pick any two elements, a and b. or . For example, if I'd pick and {1}. Then, {1}. So I think it works. I don't know if there are any other sets, A, where this works or if I'm even thinking about this correctly. (once I figure out all the sets, I'll attempt to put a proof up). Thanks in advance.