Sample Questions for Test One

1. Define the ordered pairha, bi and show that it is a set for any sets a and b. 2. Show that if {a, b} = hc, di, then either a = c and b = d, or a = d and b = c. 3. Write a formula φ(x, y) of set theory which states that y = {x}.

4. Show that the class {x : x∈ / x} is not a set. 5. Show that the intersection T C is a set for any class C. 6. Show that the intersection of a class and a set is a set.

7. Define the direct product A × B of two sets A and B and show that it is a set. 8. Define the product ΠA and show that it is a set. 9. Show that the Axiom of Replacement implies the Axiom of Comprehension

10. Show that for any sets a and b, it is not possible that a ∈ b and b ∈ a.

11. Prove by induction on n that, for any n ∈ ω and any n sets a1, . . . , an, there exists a set containing exactly those n elements. Use only the Pair and Union axioms. 12. Show that for any sets A and B, P(A ∩ B) = P(A) ∩ P(B). 13. Show that {4, 5, 6,... } is a set. 14. Define a monotone operator Γ with least fixed point P = {2n : n ∈ ω}; you may assume that addition is defined.

15. Show that for any linear ordering ≤ on a finite set A, A contains a largest element. 16. Use Tarski’s definition to show that if A and B are both finite, then the product A ∪ B is also finite. You may assume that if a set S is finite, then S ∪ {i} is finite for any i. 17. Use Tarski’s definition to prove that if A is finite, then A is not isomorphic to any proper subset of A. 18. Give the recursive definition of addition and use it to prove by induction that 0 + n = n for all n.

19. Give the recursive definition of multiplication and use it to prove by in- duction that 1 · n = n for all n. S 20. If X is a finite set and X ⊆ An, then there exists n ∈ N such that S n∈N X ⊆ i≤n Ai. 21. Show that a set A is transitive if and only if S A ⊆ A. 22. Show that the union operator is both monotone and finitary. 23. Show that if Γ is a monotone and finitary operator, then Γω is the least fixed point of Γ. 24. Show that the transitive closure trcl(a) exists for any set a. 25. Show that for any set A,(A, ∈) satisfies the Empty Set Axiom. 26. Show that if A is transitive, then (A, ∈) satisfies the Axiom of Extension- ality. 27. Let B = {1, {1}, 2}. Is B transitive? Does B satisfy Extensionality? 28. For any set A,(P(A), ∈) satisfies the Axiom of Separation 29. Show that the Empty Set Axiom follows from the Axiom of Infinity and the Axiom of Separation.

30. Show by induction that Vn ⊆ Vn+1 for all n.

31. Show that Vn is transitive for all n ∈ ω and that Vω is transitive.

32. Show that Vω satisfies the Union Axiom

33. Show that Vω satisfies the Pair Axiom

34. Show that Vω satisfies the Power Set Axiom 35. Show that the relation |x| ≤ |y| is transitive and that the relation |x| = |y| is an equivalence relation. 36. Show that for any set x, |x| < |P(x)|. 37. Show that if |x| ≤ |y|, then there is a surjection from y to x. 38. Show that a set A is infinite if and only if A is isomorphic to a proper subset of itself. 39. Show that if x is countable, then either x is finite or |x| = |ω|. 40. Show that a set x is countable if and only if it is a surjective image of ω. 41. Show that the product of two countable sets is sountable. 42. Show that the union of two countable sets is sountable. 43. Show that |P(A)| = |2A| for any set A. 44. Show that the set of finite subsets of ω is countable. 45. Show that ωn is countable for each n.

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