Bridge to Higher Mathematics, MA 1971
Exercise Set II
1. Let A and B be subsets of a universe U. Please prove the second De Morgan’s 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.
3. If U := [0, 10], A := [3, 7) and B := {3, 6, 9}, then what are A
c
U
, A
c
R
and B
c
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.
Xn
i=1
i =
n(n + 1)
2
2.
Xn
i=1
i
2 =
n(n + 1)(2n + 1)
6
6. Please find two distinct proofs that for any n ∈ Z
+, then 6 divides n
3 − n, that is,
6|(n
3 − n).
7. Suppose A and B are sets with A ⊂ B. Given the standard definition of A
c
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 not
a 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.
