Question

Prove that the union of infinitely many sets is countable using
induction.

Answer #1

Prove that tue union of countable sets is countable.

Prove the union of a finite collection of countable
sets is countable.

Prove the union of two infinite countable sets is countable.

Use the fact that “countable union of disjoint countable sets is
countable" to prove “the set of all polynomials with rational
coefficients must be countable.”

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...

A countable union of disjoint countable sets is countable
Note: countable sets can be either finite or infinite
A countable union of countable sets is countable
Note: countable sets can be either finite or infinite

Prove for each of the following:
a. Exercise A union of finitely many or countably many countable
sets is countable. (Hint: Similar)
b. Theorem: (Cantor 1874, 1891) R is uncountable.
c. Theorem: We write |R| = c the “continuum”. Then c = |P(N)| =
2א0
d. Prove the set I of irrational number is uncountable. (Hint:
Contradiction.)

SHow that a union of a finite or countable number of
sets of lebesgue measure zero is a set of lebesgue measure
zero.
Please show all steps

(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 =...

Prove that the union of two compact sets is compact using the
fact that every open cover has a finite subcover.

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