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Prove or disprove some set relations

If and prove or disprove the following:

1. .
2. .
3. .
4. .
5. .
6. .

1. 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 . ∎

2. True
Proof. This follows immediately from part (a) since . ∎
3. Not True
This is not true since and the set is not an element of (since ).
4. True
Proof. There is really nothing to prove here. By definition, is the set containing 1, so 1 is in . ∎
5. Not true
This is not true since 1 is not a set, so is not a subset of .
6. Not true
Again, 1 is not a set, so cannot be a subset of .

One comment

1. Mathieu St-Amour says:

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 .