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Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can...

Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can be written as a disjoint union in the form A ∪ B = (A \ (A ∩ B)) ∪˙ (B \ (A ∩ B)) ∪˙ (A ∩ B).

Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle.

Exercise 1.10. Prove for any set X and for any subsets A and B of X, the set A can be written as a disjoint union in the form A = (A ∩ B) ∪˙ (A ∩ B c ).

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