Mathematics 2224: Lebesgue integral
Homework exercise sheet 1
Due 3:50pm, Wednesday 2nd February 2011
1. (a) Show that if S and T are countable, then S ∪ T is countable.
(b) Deduce that the set of irrational real numbers, R \ Q, is uncountable.
2. (a) Show that N × N is countably infinite.
(b) Show that if S and T are countable, then S × T is countable.
S
(c) Show that if S1 , S2 , S3 , . . . are countable sets, then ∞
j=1 Sj is countable.
3. Let S and T be sets with the same size, meaning that there is a bijection f : S → T .
(a) Show that S is finite if and only if T is finite.
(b) Show that S is countably infinite if and only if T is countably infinite.
(c) Show that S is uncountable if and only if T is uncountable.
4. (a) Show that if a < b then the set [a, b) is uncountable.
[Hint: show that [0, 1) has the same size as [a, b).]
(b) Show that if a < b then the set (a, b) is uncountable.
(c) Deduce that any non-empty open subset of R is uncountable.
5. Let aj ∈ [0, ∞] for each j ∈ N. Show that if σ : N → N is a bijection, then
∞
X
aj =
j=1
∞
X
aσ(j) .
j=1
6. (a) Let I be an index set, and let Ai be a set for each i ∈ I. Show that if B is a
set, then
[ [
Ai ∩ B = (Ai ∩ B).
i∈I
i∈I
(b) If J is an index set and Bj is a set for each j ∈ J, show that
[ [ [ [
Ai ∩
Bj =
(Ai ∩ Bj ) .
i∈I
j∈J
i∈I
j∈J
[Remark: many of these brackets are often omitted! But I’m including them
to make things unambiguous.]
(c) Use De Morgan’s laws to establish an analogue of (b) in which intersections
and unions are swapped.