If and prove or disprove the following:

- .
- .
- .
- .
- .
- .

* True ** Proof. * To show we must show that every element in is also in . Further (since Apostol seems to distinguish between and ) we want to show that .
First, since is the only element of , and , we see that every element of is indeed contained in . Hence, .

Then, since , but , we see that . Hence , so is a proper subset of , or . ∎

* True *

* Proof. * This follows immediately from part (a) since . ∎
* Not True *

This is not true since and the set is not an element of (since ).
* True *

* Proof. * There is really nothing to prove here. By definition, is the set containing 1, so 1 is in . ∎
* Not true *

This is not true since 1 is not a set, so is not a subset of .
* Not true *

Again, 1 is not a set, so cannot be a subset of .

*Related*

Why does (b) true when A does’nt equal B ? Isin’nt that the distinction of ? I thought that either one or the other could’ve be true .