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Let A, B, C, D be sets, and consider the following: Theorem 1. A × (B...

Let A, B, C, D be sets, and consider the following:

Theorem 1. A × (B ∪ C) = (A × B) ∪ (A × C).

Theorem 2. (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).

Theorem 3. (A × B) ∆ (C × D) = (A ∆ C) × (B ∆ D).

For each, give a proof or counterexample.

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