Prove that the set of real numbers of the form e^n,n=
0,=+-1,+-2,... is countable.
Prove that the set of real numbers of the form e^n,n=
0,=+-1,+-2,... is countable.
Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of...
Prove that a countable union of countable sets countable; i.e.,
if {Ai}i∈I is a collection of sets, indexed by I ⊂ N, with each Ai
countable, then union i∈I Ai is countable. Hints: (i) Show that it
suffices to prove this for the case in which I = N and, for every i
∈ N, the set Ai is nonempty. (ii) In the case above, a result
proven in class shows that for each i ∈ N there is a...
Prove for each:
a. Proposition: If A is finite and B is countable, then A ∪...
Prove for each:
a. Proposition: If A is finite and B is countable, then A ∪ B is
countable.
b. Proposition: Every subset A ⊆ N is finite or countable.
[Similarly if A ⊆ B with B countable.]
c. Proposition: If N → A is a surjection, then A is finite or
countable. [Or if countable B → A surjection.]
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪ ... ∪ An is countable. (Hint: Induction.)
(6) Let F be the set of all functions from R to R. Show that |F|
> 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)
(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4},
{1, 4}, {1, 2, 3, 4}}, and S =...