21List of Exercises
In each exercise you may assume every proposition stated up to that point in the primer, and use it in your proof. You may not assume any proposition which comes later.
In Exercises 13 to 16, X and Y are subsets of a domain U.
Exercise 1. Prove Proposition 2 (every set is a subset of itself): Every set is a subset of itself. Exercise 2. Prove Proposition 5 (the empty set is unique), as a corollary of Propositions 1 (double inclusion) and 4 (an empty set is a subset of anything): Any two sets with no elements are identical. Exercise 3. Prove Proposition 16 (union commutes): For any sets X and Y, X ∪ Y = Y ∪ X. Exercise 4. Prove Proposition 17 (union is associative): For any sets X, Y and Z, (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z). Exercise 5. Prove Proposition 18 (uniting a set with itself): For any set X, X ∪ X = X. Exercise 6. Prove Proposition 19 (uniting with the empty set): For any set X, X ∪ ∅ = X. Exercise 7. Prove Proposition 20 (union respects inclusion): For any sets X, Y and Z, if X ⊂ Y, then X ∪ Z ⊂ Y ∪ Z. Exercise 8. Prove Proposition 24 (union distributes over intersection): For any sets X, Y and Z, X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z). Exercise 9. Prove Proposition 25 (absorption for union): For any sets X and Y, X ∪ (X ∩ Y) = X. Hint: consider applying Proposition 8 (an intersection is a subset of each set) and Proposition 22 (inclusion via union).
Exercise 10. Prove Proposition 26 (absorption for intersection): For any sets X and Y, X ∩ (X ∪ Y) = X. Hint: consider applying Proposition 14 (inclusion via intersection) and Proposition 15 (each set is a subset of the union).
Exercise 11. Prove Proposition 28 (subtracting the empty set): For any set X, X ∖ ∅ = X. Exercise 12. Prove Proposition 29 (subtracting a set from itself): For any set X, X ∖ X = ∅. Exercise 13. Prove Proposition 32 (the complement of a complement): (Xc)c = X. Exercise 14. Prove Proposition 33 (the complement of the empty set): ∅c = U. Exercise 15. Prove Proposition 34 (the complement of the domain): Uc = ∅. Exercise 16. Prove Proposition 37 (De Morgan, for union): (X ∪ Y)c = Xc ∩ Yc. Exercise 17. Prove Proposition 42 (splitting a union into disjoint pieces): For any sets X and Y, X ∪ Y = X ∪ (Y ∖ X). Exercise 18. Prove Proposition 43 (splitting a set by another set): For any sets X and Y, Y = (Y ∖ X) ∪ (X ∩ Y).