Theorem 7.2: Let S and T be two sets. Then

and

Proof: Let

Either

or

If

then

by Definition 1.4. If

then

by Definition 1.5. So

by Definition 1.1. For the reverse inclusion, let

Then by Definition 1.3, either

or

If

then

by Definition 1.4. If

then

by Definition 1.5. In either event,

so

Since we have inclusion both ways,

To see that the two sets are disjoint, assume

Then

by Definition 1.4, and

by Definition 1.5, which is a contradiction.