Is the inclusion relation on the power set of `X` an1) Equivalence relation 2) Partial order 3) Total order If `A,B` are elements of the power set of `X` and we define the relation...

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` ?

Expert Answers
degeneratecircle eNotes educator| Certified Educator

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.

To see that it's not a total order, just note that if `a,b` are two distinct elements of `X,` then `{a}` is not a subset of `{b}` and `{b}` is not a subset of ` ``{a},` so it is not the case that either `AcongB` or `BcongA.` It is only a total order if `X` has less than two elements.