Exercise Set II

1. Let A and B be subsets of a universe U. Please prove the second De Morgans law:

(A B)^c = A^c B^c

2. Prove that if A, B and C are sets, and if A B and B C, then A C.

c

3. If U := [0,10], A := [3,7) and B := {3,6,9}, then what areand B_U ?
4. Let A and B be sets. Please prove or disprove:

P(A B) = P(A) P(B)

Hint: Counterexample 5. Prove that for each n Z⁺,

1. 2.
2. Please find two distinct proofs that for any n Z⁺, then 6 divides n³ n, that is, 6|(n³ n).

c

7. Suppose A and B are sets with A B. Given the standard definition of A_B, use the axioms to show that this complement exists.
8. In terms of axiomatic set theory, please explain why a set containing all sets is nota set.
9. Is the same as {}? Explain why or why not. Hint: Cardinality.
10. Please construct on the basis of the axioms a set containing exactly three elements.