Is the inclusion relation on the power set of `X` an
1) Equivalence relation 2) Partial order 3) Total order
If `A,B` are elements of the power set of `X` and we define the relation `cong`` ` as `AcongB` if and only if `AsubseteqB` , what can we say about `cong` ?
The relation described is a partial order, but not a total order or an equivalence relation.
It is reflexive, since for any set `A` it is true that `AsubseteqA.`
It's also transitive, since `AsubseteqB` and `BsubseteqC` implies that all elements of `A` are also in `B,` which are themselves contained in `C,` so indeed every element of `A` is also an element of `C` and we have `AsubseteqC.`
However, `cong` is not symmetric in general because `AsubseteqB` doesn't imply that `BsubseteqA` (unless `X` is the empty set, which is the only way for symmetry to hold). Therefore `cong` can't be an equivalence relation.
Now we have to check for antisymmetry, which says that `AcongB` and `BcongA` are both true only if `A=B.` In the language of sets, this just says that `AsubseteqB` and `BsubseteqA` implies `A=B,` which is true.
Since it is reflexive, transitive, and antisymmetric, `cong` is a partial order.