Theorem 2.8: (The Distributivity of Cartesian Products Across Unions). Let A, B, and C be three sets. If

and

Proof: Let

Then

by Definition 1.9.

If

then

by Theorem 1.2.

If

then

In either case,

Conclude that

by Definition 1.1. For the converse inclusion, let

If

then

by Theorem 1.2, and

by Definition 1.9, so

If on the other hand,

then

by Theorem 1.2, and

so

Conclude that

Since we have containment both ways,

by Definition 1.2.

(Left Distributivity) Let

Then

If

then

If

then

In either case,

Conclude that

For the converse containment, let

If

then by Definition 1.9,

and

by Theorem 1.2, so

If on the other hand,

then

and

so

Conclude that

Since we have containment both ways,

by Definition 1.2